On the polar cap cascade pair multiplicity of young pulsars
DDraft version March 28, 2018
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ON THE POLAR CAP CASCADE PAIR MULTIPLICITY OF YOUNG PULSARS
A. N. Timokhin , and A. K. Harding Astrophysics Science Division, NASA/Goddard Space Flight Center, Greenbelt, MD 20771, USA University of Maryland, College Park (UMDCP/CRESST), College Park, MD 20742, USA (Dated: Received ; accepted ; published)
Draft version March 28, 2018
ABSTRACTWe study the efficiency of pair production in polar caps of young pulsars under a variety of conditionsto estimate the maximum possible multiplicity of pair plasma in pulsar magnetospheres. We developa semi-analytic model for calculation of cascade multiplicity which allows efficient exploration of theparameter space and corroborate it with direct numerical simulations. Pair creation processes areconsidered separately from particle acceleration in order to assess different factors affecting cascadeefficiency, with acceleration of primary particles described by recent self-consistent non-stationarymodel of pair cascades. We argue that the most efficient cascades operate in the curvature ra-diation/synchrotron regime, the maximum multiplicity of pair plasma in pulsar magnetospheres is ∼ few × . The multiplicity of pair plasma in magnetospheres of young energetic pulsars weaklydepends on the strength of the magnetic field and the radius of curvature of magnetic field lines andhas a stronger dependence on pulsar inclination angle. This result questions assumptions about veryhigh pair plasma multiplicity in theories of pulsar wind nebulae. Keywords: acceleration of particles — plasmas — pulsars: general — stars: neutron INTRODUCTIONThe idea that production of electron-positron pairs inmagnetospheres of rotation-powered pulsars is intimatelyconnected with their activity had been proposed by Stur-rock (1971) only a few years after the discovery of pulsars(Hewish et al. 1968). Since then it has become an inte-gral part of the standard pulsar model and today, thereis little doubt that an active rotationally powered pulsarproduces electron-positron plasma. Although the pulsaremission mechanism(s) is still not yet identified, there isstrong empirical evidence that pulsars stop emitting ra-dio waves when pair formation ceases – the threshold forpair formation roughly corresponds to the “death line”in pulsar parameter space, what was already noted bySturrock (1971). Furthermore, the narrow peaks in manypulsar high-energy light curves (Abdo et al. 2010) requirepervasive screening of the whole magnetosphere by pairplasma, except in narrow accelerator gaps (e.g. Watterset al. 2009; Pierbattista et al. 2015). Understanding pairplasma generation in pulsar magnetospheres is there-fore of crucial importance for developing pulsar emissionmodels.In the standard pulsar model, initially proposed byGoldreich & Julian (1969) and Sturrock (1971), the mag-netosphere is filled with dense pair plasma which screensthe accelerating electric field everywhere except somesmall zones which are responsible for particle accelera-tion and emission. Pair plasma is primarily created viaconversion of γ rays in the strong magnetic field near thepolar caps (PCs). Pair production in the PCs is a “cor-nerstone” of the standard model – without dense plasmaproduced at the PCs, at the base of open magnetic fieldlines, the magnetosphere would have large volumes withunscreened electric field, as pair creation in e.g. outergaps (Cheng et al. 1976) cannot screen the electric field [email protected] over the rest of the magnetosphere.Charge starvation (Arons & Scharlemann 1979) or vac-uum gaps (Ruderman & Sutherland 1975) at the polarcap, when the number density of charged particles is notenough to screen the electric field, leads to formationof accelerating zone(s). Some charged particles enterthis zone, are accelerated to very high energies and emitgamma-rays which are absorbed in the ultrastrong mag-netic field, creating electron-positron pairs. The pairs,being relativistic, can also emit pair producing photonsand so the avalanche develops until photons emitted bythe last generation of pairs can no longer produce pairsand escape the magnetosphere.The pair plasma created by pulsars flows out of themagnetosphere along open magnetic field lines and pro-vides the radiating particles for the surrounding PulsarWind Nebulae (PWNe). Models of PWNe depend (atleast) on the density of the plasma, what produces theobserved synchrotron and inverse Compton emission. Es-timates of the pair multiplicity (the number of pairs pro-duced by each primary accelerated particle) needed toaccount for the emission from the Crab pulsar wind neb-ula (PWN) range from about 10 − (de Jager et al.1996) up to 10 (e.g. Bucciantini et al. 2011); for the VelaPWN the multiplicity is estimated to be about 10 (deJager 2007). PWNe therefore give the most compellingevidence for pair production and pair cascades in at leastyoung energetic pulsar magnetospheres.Although PWNe are observed only around young pul-sars ( < a few times 10 yrs), evidence for pairs, at leastfor high plasma densities larger than those provided byprimary particles, can also be found in older pulsars.Synchrotron absorption models for the eclipse in the dou-ble pulsar system PSR J0737-3039 (Arons et al. 2005;Lyutikov 2004) require a pair multiplicity of around 10 for the recycled 22 ms pulsar in that system.The cascade process in pulsar polar caps has been a r X i v : . [ a s t r o - ph . H E ] M a r Timokhin & Harding the subject of extensive numerical as well as analyticalstudies (e.g. Daugherty & Harding 1982; Gurevich & Is-tomin 1985; Zhang & Harding 2000; Hibschman & Arons2001a,b; Medin & Lai 2010). The pair plasma multi-plicity obtained in these studies was significantly lowerthan estimates of pair plasma multiplicity in PWNe, asit did not exceed ∼ few × . Most of those works con-sidered pair creation together with the particle accel-eration which makes these analyses dependent on theacceleration model considered. These studies also as-sumed steady, time-independent acceleration of the pri-mary particles. However, recent studies by Timokhin(2010); Timokhin & Arons (2013) have found that pulsarpolar cap pair cascades are not time-steady in the gen-eral case of arbitrary current, particularly those requiredby global magnetosphere models (e.g Contopoulos et al.1999; Timokhin 2006; Spitkovsky 2006; Kalapotharakos& Contopoulos 2009).In this paper we study the question of what is the max-imum pair multiplicity achievable in pulsar polar cap cas-cades and under which circumstances is it achieved. Incontrast to previous pair cascade studies, we take a mul-tistep approach. We consider the physical processes inpair cascades and particle acceleration models separatelyin order to clearly set apart different factors influencingthe efficiency of pair cascades. We first assess how eachof the microscopic processes affects the final multiplic-ity and the pulsar parameter ranges that result in thelargest possible pair multiplicity. Then, we employ themost recent model of non steady-state particle accelera-tion in pulsar polar caps and derive a simple analyticalestimate for the maximum energy of particles acceleratedin a non-stationary cascade. One on the most importantresults of our study is a strong upper limit on pair plasmamultiplicity in pulsars.We limit ourselves to the case of cascades at the polarcaps of young pulsars as from previous theoretical stud-ies of polar cap cascades, such pulsars are expected tobe the most efficient pair producers. We rely on resultsof previous cascade studies in our choice of the specificcascade process, namely cascades initiated by curvatureradiation of primary particles.The plan of the paper is as follows. In § §§ § Pulsars with strong polar cap cascades should have potentialdrop over the polar cap well in excess of the pair formation thresh-old as well as large magnetic fields B (cid:38) G and short rotationalperiods, so that particle acceleration happens over a short distanceand cascade develops in the region with strong magnetic field. Thebest single parameter selecting pulsars with such properties is thesmall characteristic age τ = P/ P . A detailed discussion of thepulsar parameter range where approximations used in this paperare formally applicable is given at the end of § We discuss uncertainties of current pulsar models in § PHYSICS OF POLAR CAP CASCADES: ANOVERVIEWAn electron-positron cascade can be thought of as aprocess of splitting the energy of primary particles intothe energies of secondary particles. The maximum mul-tiplicity, the number of secondary particles for each pri-mary particle, of an ideal cascade initiated by a singleprimary particle with energy (cid:15) p would be κ max (cid:39) (cid:15) p (cid:15) γ, esc . (1)where (cid:15) γ, esc is the maximum energy of photons escapingfrom the cascade (or the minimum energy of pair pro-ducing photons). Not all of the primary particle energygoes into pair producing photons, and pairs created inthe cascade do not radiate all of their energy into thenext generation of pair producing photons. Hence, κ max is only the theoretical upper limit on the multiplicity ofa real cascade. The total pair yield of the cascade is acombination of four factors: (a) number of primary par-ticles, (b) initial energy of primary particle – the higherthe energy the more pairs can be produced, (c) thresholdfor pair formation – the lower the threshold the higherthe multiplicity, (d) efficiency of splitting the energy ofprimary particles into pairs – the higher the fraction ofparticle energy going into pair creating photons, as op-posed to the final kinetic energy of the particles and pho-tons below the pair formation threshold, the higher themultiplicity.In young, fast rotating pulsars, the electric field in thepolar cap acceleration zones is strong and primary par-ticles can be accelerated up to very high Lorentz factors, γ (cid:38) . At these energies the most effective radiationprocess is curvature radiation (CR). CR efficiency growsrapidly with the particle energy and for young pulsarsbecomes the dominant emission mechanism for primaryparticles. For secondary particles, which are substan-tially less energetic than the primary ones, the primaryway to create pair producing photons is via synchrotronradiation. In § (cid:38) × G,that channel never becomes the dominant one for pairproduction in young pulsars, at best resulting in pairmultiplicity comparable to the one of the synchrotronchannel. Hence, in young pulsars which are expected tohave the highest multiplicity pair plasmas, the polar capcascades should operate primarily in the CR-synchrotronregime – all known studies of cascades in pulsar polar capagree on this point. In this paper we study in detail CR-synchrotron cascades. The resulting multiplicities willbe good estimates for a wide range of pulsar parame-ters, however, as we consider only synchrotron radiationof secondary particles, for pulsars with magnetic field (cid:38) × G our analysis might underestimate the cas-cade multiplicity by a factor of ∼
2, see § air multiplicity in young pulsars neutron star pair creationsynchrotron photonCR photon generation number Figure 1.
Schematic representation of electron-positron cascadein the polar cap of a young pulsar, see text for description. cascade development in polar cap regions of young pul-sars. Shown are the first two generations in a cascadeinitiated by a primary electron. Primary electrons emitCR photons (almost) tangent to the magnetic field lines;primary electrons and CR photons are generation 0 par-ticles in our notation. Magnetic field lines are curvedand the angle between the photon momentum and themagnetic field grows as the photon propagates furtherfrom the emission point. When this angle becomes largeenough, photons are absorbed and each photon createsan electron-positron pair – generation 1 electron andpositron. The pair momentum is directed along the mo-mentum of the parent photon and at the moment of cre-ation, the particles have non-zero momentum perpendic-ular to the magnetic field. They radiate this perpendic-ular momentum almost instantaneously via synchrotronradiation and then move along magnetic field lines. Al-though these secondary particles are relativistic, theirenergy is much lower than that of the primary electronand their curvature photons cannot create pairs. Afterthe emission of synchrotron photons, secondary particles(generation 1 and higher) no longer contribute to cas-cade development. Generation 1 photons (synchrotronphotons produced by the generation 1 particles) are alsoemitted (almost) tangent to the magnetic field line – asthe secondary particles are relativistic – and propagatesome distance before acquiring the necessary angle to the magnetic field and creating generation 2 pairs. Thesepairs in their turn radiate their perpendicular momentumvia synchrotron radiation, emitting generation 2 photons.The cascade initiated by a single CR photon stops at ageneration where the energy of synchrotron photons fallsbelow (cid:15) γ, esc .Only primary particles emit pair producing photonsas they move along the field lines; all secondary par-ticles emit pair producing photons just after their cre-ation. The cascade development can be thus dividedinto two parts: (i) primary particles emit CR photonsas they move along magnetic field lines and (ii) each CRphoton gives rise to a synchrotron cascade, when syn-chrotron photons create a successive generation of pairswhich emit the next generation synchrotron photons atthe moment of creation. This division goes between gen-eration 0 and all subsequent cascade generations.In the following sections we analyze all four factorsregulating the yield of CR-synchrotron cascades listed atthe beginning of this section (in reverse order, from d toa). In §§ §
8. We start with gamma-ray absorption in a strongmagnetic field in §
3; then we discuss the efficiency of thesynchrotron cascade in §
4. Curvature radiation is consid-ered in § § § §§ § § § §
10, we address item a – the mean flux of primary par-ticles and the total yield of a CR-synchrotron cascade inan energetic pulsar. We argue that this is the most im-portant factor regulating pair yield in energetic pulsars.Despite the uncertainty in determining the mean flux ofprimary particles, we can set a rather strict upper limiton pair multiplicity in pulsars. PHOTON ABSORPTION IN THE MAGNETICFIELD3.1.
Opacity for γ − B pair production The opacity for single photon pair production in strongmagnetic field is (Erber 1966) α B ( (cid:15) γ , ψ ) = 0 . α f λ c b sin ψ exp (cid:18) − χ (cid:19) (2)where b ≡ B/B q is the local magnetic field strength B normalized to the critical quantum magnetic field B q = e/α f λ c = 4 . × G, ψ is the angle betweenthe photon momentum and the local magnetic field, α f = e / (cid:126) c ≈ /
137 is the fine structure constant, and λ c = (cid:126) /mc = 3 . × − cm is the reduced Compton Timokhin & Harding
124 68 1012141618 log ϵ γ l og B ρ c = cm
246 8101214161820 log ϵ γ ρ c = cm
468 101214161820 log ϵ γ ρ c = cm Figure 2.
Contour plot of 1 /χ a as a function of the logarithms of the magnetic field strength B in Gauss, and photon energy (cid:15) γ normalizedto the electron rest energy, for three values of the radius of curvature of magnetic field lines ρ c = 10 , , cm. 1 χ a values shown onthis plot are calculated from eq. (7) and are not corrected for the kinematic threshold (see text).
124 68 1012 141618 log ϵ γ l og B ρ c = cm
24 68 1012 14161820 log ϵ γ ρ c = cm
46 810 1214 161820 log ϵ γ ρ c = cm Figure 3.
Contour plot of 1 /χ a corrected for kinematic threshold according to eq. (9). These values of 1 /χ a are used in all calculationswithin semi-analytic model. Notations are the same as in Fig. 2. wavelength. The parameter χ is defined as χ ≡ (cid:15) γ b sin ψ , (3)where (cid:15) γ is the photon energy in units of m e c . For con-venience from here on, all particle and photon energieswill be quoted in terms of m e c . The optical depths forpair creation by a high energy photon in a strong mag-netic field after propagating distance l is τ ( (cid:15) γ , l ) = (cid:90) l α B ( (cid:15) γ , ψ ( x )) dx , (4)where integration is along the photon’s trajectory.Expression (2) is accurate if the magnetic field is smallcompared to the critical field B q , b < . (cid:15) γ sin ψ > B q /
3. In this paper we study cascades inpulsars with “normal” magnetic fields and so we neglecthigh-field effects.Expression (2) becomes inaccurate when pairs are cre-ated at low Landau levels, near the pair formation thresh-old, it overestimates the opacity and even formally allowspair formation below the kinematic threshold (cid:15) γ sin ψ =2. However, pairs created at low Landau levels will notgive rise to strong cascades, as their perpendicular en-ergy will be too low to emit pair-producing synchrotronphotons. Hence, for the case of strong cascades, accuratetreatment of pair formation near the kinematic thresholdis not necessary. Throughout this paper we use Erber’sapproximation (2) in all our analytical calculations butintroduce a cut off at the kinematic threshold (cid:15) γ sin ψ = 2as described at the end of this section, thus taking intoaccount the cessation of pair formation below the the air multiplicity in young pulsars l γ is comparable to or larger than the characteristicscale of the magnetic field variation L B , this photon willnot initiate a strong cascade with the same emission pro-cesses by which it was produced. The reason for this isas follows. The opacity for γB pair production exponen-tially depends on the magnetic field strength and photonenergy via χ . The energy of the next generation photonwill be smaller than that of the primary one, and, be-cause the primary photon has already traveled the dis-tance over which the magnetic field has substantially de-clined, the magnetic field along the next generation pho-ton’s trajectory will be substantially weaker than thatalong the primary photon’s trajectory . The next gener-ation photon’s mfp will be much larger than than that ofthe primary photon, and, even if this secondary photonwill be absorbed, it will be the last cascade generation.Hence, in a strong cascade, for all but the last gener-ation photons, l γ (cid:28) L B . A reasonable estimate for L B would be the distance of the order of the NS radius R ns asany global NS magnetic field decays with the distance as( r/R ns ) − δ , δ ≥
3. Very localized, sun spot like magneticfields, are in our opinion of no importance for the gen-eral pulsar case as the probability of such a “spot” to lieat the polar cap should be rather low, i.e. most pulsarsshould be able to produce plasma in a more or less regularmagnetic field. A dipole field, δ = 3, is often consideredas a reasonable assumption for a general pulsar model.Pure dipole field, however, seems to be a too idealizedapproximation, as even if the NS field is a pure dipole, itwill be slightly disturbed by the currents flowing in themagnetosphere. In general, near dipole magnetic fieldswith different curvatures of magnetic field lines shouldbe examined in cascade models.We consider strong cascades with large multiplicities,where, as argued above, photons propagate distancesmuch shorter than the characteristic scale of the mag-netic field variation, so we assume that in the regionwhere most of the pairs are produced the magnetic fieldis constant. The radius of curvature of magnetic fieldlines ρ c is not smaller than L B , and as l γ (cid:28) L B theangle ψ is always small, the approximation sin ψ ≈ ψ isvery accurate. For photons emitted tangent to the mag-netic field line, dx = ρ c dψ . In our approximation both b and ρ c are constants. From eq. (3) we have ψ = 2 χ/(cid:15) γ b ,and substituting it into eq. (4) we can express the opticaldepth τ to pair production as an integral over χτ ( (cid:15) γ , l ) = A τ ρ c (cid:15) γ b (cid:90) χ ( (cid:15) γ ,ψ ( l ))0 χ exp (cid:18) − χ (cid:19) dχ , (5)where A τ ≡ . α f /λ c ≈ . × cm − .Integrating eq. (5) over χ by parts two times we canget an expression for τ in terms of elementary functionsand the exponential integral function Ei: τ ( χ ) = A τ ρ c (cid:15) γ b × In the polar cap, photons are emitted by ultrarelativistic parti-cles moving along magnetic field lines. At emission points photonsare almost tangent to field lines and the differences in initial photonpitch angles for different generations can be neglected. (cid:20) χ (cid:18) − χ (cid:19) e − χ −
89 Ei (cid:18) − χ (cid:19)(cid:21) . (6)Ei( z ) is a widely used special function, defined as Ei( z ) = − (cid:82) ∞− z exp( − t ) /t dt . There are efficient numerical algo-rithms for its calculation implemented in many numer-ical libraries and scientific software tools; using eq. (6)for calculation of the optical depths will result in muchmore efficient numerical codes than direct integration ofeq. (5).As the optical depth to pair formation grows exponen-tially with χ (and distance), for analytical estimates it isreasonable to assume that all photons are absorbed whenthey have traveled the distance l γ such that τ ( l γ ) = 1.We denote the value of χ when the optical depth reaches1 as χ a : χ a : τ ( χ a ) = 1 . (7)In all our computations we will use χ a as the value of theparameter χ at the point of the photon’s absorption. χ a is a solution of the non-linear equation (7) with τ givenby eq. (6). Because of the exponential dependence of τ on 1 /χ it is to be expected that 1 /χ a should have a closeto linear dependence on logarithms of (cid:15) γ , b , and ρ c . Wesolved equation (7) numerically for different values of (cid:15) γ , b , and ρ c and, indeed, the inverse quantity 1 /χ a dependsalmost linearly on log (cid:15) γ , log b , and log ρ c in a wide rangeof these parameters. Making a modest size table of 1 /χ a values one can later use piecewise interpolation to find aparticular value of χ a quite accurately.In Fig. 2 we plot contours of 1 /χ a as functions oflog( (cid:15) γ ) and log( B ) for three different values for the ra-dius of curvature of the magnetic field lines. We want topoint out that 1 /χ a differs from the value 1 /χ a = 15 usedby Ruderman & Sutherland (1975), especially for higherenergy photons. Although this difference is only a factorof a few, as we will point out later, the cascade efficiencyin each generation depends on the corresponding value of χ a , and in a strong cascade with several generations, theestimate for the final multiplicity will be substantiallyaffected by the value of χ a .Erber’s expression is not applicable for pairs producedat low Landau levels as it overestimates the opacity and,formally, solutions for χ a obtained from eq. (7) allowspair creation even for (cid:15) γ sin ψ <
2. In our analyti-cal treatment we introduce a limit on χ a to correct forthe kinematic threshold. For photons above kinematicthreshold from eq. (3) it follows that χ a > b , (8)In all our analytical calculation we get ˜ χ a from eq. (7)and then use the maximum value of ˜ χ a and b : χ a = max ( ˜ χ a , b ) . (9)In Fig. 3 we plot contours of 1 /χ a which incorporate cor-rections to χ a due to the kinematic threshold accordingto eq. (9). It is clear from the plot that this correction af-fects only cases with high magnetic field and low particleenergies. 3.2. Energy of escaping photons
As discussed above, photons escaping the cascade arethose whose mfp l γ is larger than the distance of signifi-cant magnetic field attenuation L B . The formal criteria Timokhin & Harding .5 11.5 22.53 3.5 log ρ c l og B log ϵ γ ,esc Figure 4.
Energy of escaping photons: contours of log (cid:15) γ, esc as afunction of logarithms of the radius of curvature of magnetic fieldlines ρ c in cm and magnetic field strength B in Gauss for s esc = 1. we use for calculating the energy of escaping photons (cid:15) γ, esc is l γ ( (cid:15) γ, esc ) = s esc R ns ; s esc is a dimensionless pa-rameter quantifying the escaping distance in units of R ns .The photon mfp is l γ = ρ c ψ a and expressing the anglebetween the photon momentum and the magnetic fieldat the point of absorption ψ a through χ a we get a (non-linear) equation for (cid:15) γ, esc (cid:15) γ, esc = 2 ρ c s esc R ns χ a ( (cid:15) γ, esc , b, ρ c ) b . (10)The non-linearity in this equation is due to dependenceof χ a on (cid:15) γ, esc , b , and ρ c . Using an interpolation tablefor 1 /χ a this equation is very easy to solve numericallyfor all reasonable values of physical parameters involved.In Fig. 4 we plot energy of escaping photons, log (cid:15) γ, esc as a function of the radius of curvature of magnetic fieldlines ρ c and magnetic field strength B for s esc = 1. Forsmaller values of s esc the whole plot would move to theleft. This figure shows (an obvious) trend that for highermagnetic field and smaller radii of curvature, the energyof escaping photons is lower, which allows for more cas-cade generations and larger multiplicity. The break incontour lines around log B (cid:39) . ψ along photon’s travel path by expanding the integral ineq. (4) around its upper endpoint, as was done by Hib-schman & Arons (2001b). However here we try to exploredifferent possible magnetic field configuration – explor-ing parameter space in ρ c – and our simple estimate isaccurate to a factor of a few. SYNCHROTRON CASCADE In this section we discuss the synchrotron cascade,where most of the electron-positron pairs are created.The synchrotron cascade is the part of the whole cas-cade that is initiated by generation 0 photons. In thesynchrotron cascade each generation’s primary photon isdivided into many (lower energy) next generation’s pair-producing photons by synchrotron radiation of freshlycreated pairs.4.1.
Fraction of parent photon energy remainingin the cascade
A high energy photon, when absorbed in the magneticfield, produces an electron and a positron; the total en-ergy of these particles is equal to the energy of the pho-ton. Particle momenta just after production have pitchangles equal to ψ a , when pair production takes place wellabove threshold. When pairs are created at high Landaulevels, as is the case in strong cascades, relativistic par-ticles have non-zero pitch angles and they radiate theirperpendicular energy via synchrotron radiation; in su-perstrong magnetic fields, this happens almost instanta-neously. The component of particle momentum parallelto the magnetic field is unaffected by synchrotron radi-ation and so the final Lorentz factor of the particle (cid:15) ± , f will be (cid:15) ± , f = (1 − β (cid:107) ) − / ≈ (cid:15) ± , i (cid:2) ψ a (cid:15) ± , i ) (cid:3) − / (11)where β (cid:107) ≡ v (cid:107) /c is particle velocity along the magneticfield line and (cid:15) ± , i is the initial Lorentz factor of the par-ticle right after creation. If the photon absorption hap-pens at χ a < § ψ a in eq. (11) through χ a , ψ a = 2 χ a /(cid:15) γ b , and the initial particle energy through thephoton energy (cid:15) γ , (cid:15) ± , i = (cid:15) γ / (cid:15) ± , f as a functionof χ a and b (cid:15) ± , f = (cid:15) γ (cid:20) (cid:16) χ a b (cid:17) (cid:21) − / . (12)The fraction ζ syn of photon energy radiated as syn-chrotron photons, which is going into subsequent paircreation, is ζ syn = 2( (cid:15) ± , i − (cid:15) ± , f ) (cid:15) γ = 1 − (cid:20) (cid:16) χ a b (cid:17) (cid:21) − / . (13)Because of the kinematic threshold (8) the minimumvalue of this fraction is ζ syn | χ a = b (cid:39) . at least (cid:39)
30% of absorbed photon energy will go intosynchrotron radiation of created pairs. In Fig. 5 we plotthe fraction of pair-producing photon energy radiated assynchrotron photons by freshly created pairs given byeq. (13). Contours of ζ syn are plotted as functions of thephoton energy (cid:15) γ and magnetic field strength B , the ra-dius of curvature was assumed to be ρ c = 10 cm. Thedependence of ζ syn on ρ c (via χ a ) is very weak, and Fig. 5 as we mentioned above, the physics of near-threshold pair for-mation is more complicated and our simplified treatment is lessaccurate in this regime. air multiplicity in young pulsars .3.6 .8 .9.95 .98 log ϵ γ l og B ζ syn Figure 5.
Fraction of the parent photon energy radiated as syn-chrotron photons by freshly created pairs: contours of ζ syn as afunction of logarithms of the parent photon energy (cid:15) γ and mag-netic field strength B in Gauss for ρ c = 10 cm. is a good representation of how ζ syn depends on (cid:15) γ and B for any ρ c of interest.It is evident from Fig. 5 that for higher magnetic fieldstrengths, B (cid:38) × G, and lower energies of parentphotons, a progressively smaller fraction of the parentphoton energy goes into synchrotron photons; the restremains in the kinetic energy of the created pairs mov-ing along magnetic field lines. The portion of the parentphoton energy energy left in kinetic energy of pairs doesnot go into production of next generation pairs but is“lost” from the synchrotron cascade . The reason forthis is that for higher magnetic field strengths pairs arecreated when the photon has a smaller pitch angle ψ a ,so that a smaller fraction of the photon energy goes intoperpendicular pair energy, and hence, a smaller fractionof the photon energy is emitted and remains in the cas-cade. 4.2. Number of secondary photons
At each pair creation event the parent photon is ef-fectively transformed into an electron-positron pair andlower energy synchrotron photons. Those photons be-come parent photons for the next cascade generation orescape the magnetosphere, terminating the cascade.The characteristic energy of synchrotron photons emit-ted by a newly created particle in terms of quantities usedin this paper is given by (cid:15) γ, syn = 32 b ψ a (cid:15) ± , i . (14)The number of synchrotron photons with the character-istic energy (cid:15) γ, syn – these photons carry most of the en-ergy of synchrotron radiation – emitted at each event the kinetic energy of pairs might be tapped by RICS cascadebranches, see §
24 68 10152025 log ϵ γ l og B n syn Figure 6.
Number of synchrotron photons with the characteristicenergy (cid:15) γ, syn emitted at each γ → e + e − conversion: contours of n syn as a function of logarithms of the parent photon energy (cid:15) γ and magnetic field strength B in Gauss for ρ c = 10 cm. of conversion of a parent photon with energy (cid:15) γ into anelectron-positron pair is n syn (cid:39) ζ syn (cid:15) γ (cid:15) γ, syn = 43 ζ syn χ a . (15)In eq. (15) both ζ syn and χ a are functions of B , (cid:15) γ , and ρ c . In Fig. 6 we plot contours of n syn as functions of thephoton energy and magnetic field strength, the radiusof curvature was assumed to be ρ c = 10 cm. Two cleartrends are visible on this plot: the lower the energy of theprimary photon the larger the number of secondary syn-chrotron photons produced at each conversion event, andthe higher the magnetic field the smaller is the numberof synchrotron photons. The first one is a general trendof emission processes when higher energy particles emitless photons which, however, have larger energies. Thesecond trend is due to the suppression of the synchrotroncascade discussed above, in § Multiplicity of synchrotron cascade
The generation i + 1 cascade photon is a synchrotronphoton emitted at the event of pair creation by a photonof generation i . Expressing b , ψ a and (cid:15) ( i ) γ through χ a and (cid:15) γ from eq. (14) we get for the characteristic energy ofthe next generation photon (cid:15) ( i +1) γ = 34 χ a (cid:15) ( i ) γ , (16)where (cid:15) ( i ) γ and (cid:15) ( i +1) γ are energies of i ’th and ( i + 1)’thgeneration photons. The photon energy degrades witheach successive generation of the cascade. This degrada-tion accelerates as the cascade proceeds through genera-tions because with the decrease of the photon energy χ a increases, see Fig. 2. Timokhin & Harding
The photon mfp in a constant magnetic field goes as l γ ∝ χ a /(cid:15) γ . For photon energies (cid:15) γ (cid:46) , when χ a (cid:38) (cid:15) ( i +1) γ ≥ (cid:15) γ, esc . The total number of particles gener-ated in synchrotron cascades initiated by a primary pho-ton can be calculated by summation over all generationsof the number of pairs produced in each cascade gen-eration. The algorithm for this calculation is shown inAppendix A, algorithm 1. We will use this algorithmfor calculation of the polar cap cascade multiplicity afterwe discuss curvature radiation, the radiation mechanismresponsible for generating the primary photons for syn-chrotron cascades in young energetic pulsars.Finally we wish to point out how the magnetic fieldstrength affects the multiplicity of the synchrotron cas-cade. From discussions presented in §§ CURVATURE RADIATIONIn this section we discuss how primary particles emitphotons which “launch” the synchrotron cascade. As wediscussed above the most efficient process for supplyingthe primary (generation 0) photons in young energeticpulsars is curvature radiation.5.1.
Fraction of the primary particle energy going intothe cascade
Ultrarelativistic particles moving along curved mag-netic field lines emit electromagnetic radiation withpower (Jackson 1975) P cr = 23 e m e c (cid:18) c ρ c (cid:19) (cid:15) ± , (17)where P cr is normalized to m e c / sec, (cid:15) ± is the particle’senergy normalized to m e c ; we do not distinguish be-tween electrons and positrons. Since we are treating thepair generation problem separately from the problem ofparticle acceleration, we consider cascades produced byparticles injected in a region with screened electric field,so that particles are not accelerated and only lose en- .001.01 .1 .25.5 .75 .9.95 log ϵ ± l og ρ c ζ CR Figure 7.
Fraction of the primary particle energy emitted as CRphotons over the distance s cr = 1: contours of ζ cr as a functionof logarithms of the initial particle energy (cid:15) ± and the radius ofcurvature of magnetic field lines ρ c in cm. ergy to CR. The particle’s energy decreases with timeaccording to the equation of motion d(cid:15) ± dt = − P cr . (18)Solving eq. (18) we get for the particle energy after ittravels the distance s from the injection point (cid:15) ± ( s ) = (cid:15) ± (cid:20) H ( (cid:15) ± ) ρ s (cid:21) − / (19)(see also Harding 1981), where s is normalized to R ns , (cid:15) ± is the initial particle energy; constant H is defined as H = (2 / R ns r e ≈ . × − cm , where r e = e /m e c is the classical electron radius.The fraction ζ cr of the initial particle energy lost tocurvature radiation after a particle has traveled distance s cr is ζ cr ( s cr ) = 1 − (cid:15) ± ( s cr ) (cid:15) ± . (20)If the energy of these CR photons goes into creation ofelectron-positron pairs, ζ cr gives the efficiency of the CRpart of the full cascade. The electric field in the ac-celeration zone transforms electromagnetic energy intoparticle’s kinetic energy, which is then radiated as pairproducing photons. Only the photon’s energy can be di-vided in chunks carried by a large number of pairs. Thecascade will have high efficiency if (i) primary particleshave high energy, (ii) emit most of their energy as pho-tons, and (iii) inject these photons in the region wherethe synchrotron cascade can work effectively, i.e. in aregion close to the NS which is smaller than the charac-teristic scale of magnetic field variation L B .In Fig. 7 we plot the fraction of the primary particleenergy emitted as CR photons after the particle has trav- air multiplicity in young pulsars log ρ c l og B log ϵ ± ,th Figure 8.
Critical particle energy above which it can emit pair-producing photons via curvature radiation: contours of log (cid:15) ± , th as a function of logarithms of the radius of curvature of magneticfield lines ρ c in cm and magnetic field strength B in Gauss for s esc = 1. eled distance s cr = 1. Shown are contours of ζ cr ( s cr ) asa function of the initial particle energy (cid:15) ± and the radiusof curvature of magnetic field lines ρ c . For smaller val-ues of s cr the whole plot would move to the right. CR ismost efficient in transferring particle energy into the cas-cade in the parameter space corresponding to the lowerright triangular region of Fig. 7. Going from the upperleft (smaller (cid:15) ± , larger ρ c ) to the lower right (larger (cid:15) ± ,smaller ρ c ) on this plot, not only the particle energy in-creases but also the fraction of the energy which can gointo the cascade.For a certain range of (cid:15) ± and ρ c the energy put into thecascade by the primary particle grows faster than the en-ergy of that particle, i.e. the fraction of particle’s energygoing into the cascade increases stronger than linearlywith the energy of the particle. For more or less regularglobal magnetic fields, with ρ c (cid:38) cm, the transitionbetween effective and ineffective CR cascades occurs atparticle energies (cid:15) ± ∼ , and the efficiency is very sen-sitive to the particle energy. In this parameter rangeeven a modest increase of the primary particle energycan result in a large boost of the cascade multiplicity.5.2. Energy of CR photons and critical particle energy
The characteristic energy of CR photons emitted by aparticle with the energy (cid:15) ± is (Jackson 1975) (cid:15) γ, cr = 32 λ c ρ c (cid:15) ± ≈ . × ρ − (cid:15) ± , , (21)where ρ c, 7 ≡ ρ c / cm and (cid:15) ± , ≡ (cid:15) ± / . The numberof CR photons emitted by the particle while travelingdistance ds normalized to R ns is d n cr ds (cid:39) R ns c P cr (cid:15) γ, cr (22) Each CR photon above the pair formation threshold willbe a primary photon for the synchrotron cascade dis-cussed in §
4. The critical energy at which primary par-ticles can produce pair-creating CR photons can be cal-culated by equating (cid:15) γ, cr given by eq. (21) to the escapephoton energy (cid:15) γ, esc from eq. (10). In Fig. 8 we plotthe critical particle energy which could initiate pair pro-duction with CR photons (cid:15) ± , th for s esc = 1. Shown arecontours of log (cid:15) ± , th as a function of the radius of curva-ture of magnetic field lines ρ c and magnetic field strength B . Primary particles should have energies (cid:15) ± (cid:38) tobe able to initiate pair production via CR.5.3. Multiplicity of CR-synchrotron cascade
The multiplicity of the the CR-synchrotron cascade –the total number of particles produced by a single pri-mary electron or positron accelerated in the gap – canbe computed by multiplying the number of CR photons n cr , eq. (22), by the number of particles produced inthe synchrotron cascade initiated by these photons n syn ,eq. (15), and integrating it over the distance where CRcan initiate a cascade κ cr -syn = (cid:90) s cr n syn d n cr ds ds . (23)The actual algorithm we use to compute the total multi-plicity is the Algorithm 2 from Appendix A. Integrationin eq. (23) is done assuming constant values for B and ρ c , as discussed in § κ cr -syn as afunction of the initial particle energy log (cid:15) ± and magneticfield strength log B for three different radii of curvatureof magnetic field lines ρ c = 10 , , cm assuming s cr = s esc = 1.It is evident from these plots that in a dipolar magneticfield, with ρ c ≈ cm, the maximum achievable multi-plicity is κ cr -syn ∼ few × even in cascades initiatedby extremely energetic primary particles. If the radiusof curvature is an order of magnitude less, a rather highmultiplicity κ cr -syn (cid:38) could be achieved in polar capcascades for magnetic field strength B ∼ G andparticle energies (cid:15) ± (cid:38) , parameters quite realistic foryoung pulsars. For strongly non-dipolar magnetic field,with ρ c ≈ cm the multiplicity can be another orderof magnitude higher κ cr -syn ∼ few × .The properties of CR do not depend on the strengthof the magnetic field, therefore the effect of the magneticfield strength on the multiplicity of CR-synchrotron cas-cade is due to the synchrotron cascade and how manyCR photons pair produce. As discussed at the end of § B ∼ G. The value of thisoptimum magnetic field grows slightly with increase of ρ c , but it stays around ∼ few × G even for dipolarmagnetic fields. This is noteworthy in view of the factthat B ∼ G is the typical value of magnetic fieldstrength for normal pulsars. For any given energy of theprimary particle the decrease of the cascade multiplicity0
Timokhin & Harding
12 33.5 44.5 55.56 6.5 log ϵ ± l og B ρ c = cm log ϵ ± ρ c = cm log ϵ ± ρ c = cm Figure 9.
Multiplicity of CR-synchrotron cascade: contours of log κ cr -syn as a function of logarithms of the primary particle energy (cid:15) ± and magnetic field strength B in Gauss for three values of the radius of curvature of magnetic field lines ρ c = 10 , , cm. Assumedvalues for characteristic lengths: s cr = s esc = 1. .001 .01 .01.1 .25 .255.5 6.0 6.5 7.0 7.5 8.011.011.512.012.513.0 log ϵ ± l og B ρ c = cm .001 .01 .01.1 .25 .255.5 6.0 6.5 7.0 7.5 8.0 log ϵ ± ρ c = cm .001 .01 .1 .25.255.5 6.0 6.5 7.0 7.5 8.0 log ϵ ± ρ c = cm Figure 10.
Efficiency of CR-synchrotron cascade as given by eq. (24): contours of ζ cr -syn as a function of logarithms of the primary particleenergy (cid:15) ± and magnetic field strength B in Gauss for three values of the radius of curvature of magnetic field lines ρ c = 10 , , cm.Assumed values for characteristic lengths: s cr = s esc = 1. towards stronger magnetic fields is faster than for weakerfields.The dependence of cascade multiplicity on the initialenergy of the primary particle is non-linear. Let us con-sider what happens when the initial particle energy (cid:15) ± goes from the highest to the lowest value (horizontal di-rection in plots of Fig. 9). For the highest values of (cid:15) ± multiplicity decreases uniformly, but then it dropsby an order of magnitude in a rather small range of (cid:15) ± (for B ∼ G it happens around (cid:15) ± ∼ . for ρ c = 10 cm, (cid:15) ± ∼ . for ρ c = 10 cm, and (cid:15) ± ∼ . for ρ c = 10 cm). After about half a decade of (cid:15) ± val-ues the multiplicity drops again to 1, when no parti-cles can be produced (for B ∼ G it happens around (cid:15) ± ∼ . for ρ c = 10 cm, (cid:15) ± ∼ . for ρ c = 10 cm,and (cid:15) ± ∼ for ρ c = 10 cm). The first effect is due to the decrease of the efficiency of CR, discussed in § ζ cr , where it becomes less than 10% (see Fig. 7) manifestsin a rapid decrease of cascade multiplicity – by an orderof magnitude – on all plots of Fig. 9 for magnetic fieldstrengths where the maximum multiplicity is achieved.This drop in κ cr -syn is most prominent for B ∼ Gand is less pronounced for both higher and lower mag-netic field strengths due to lower efficiency of the cascadediscussed above. The second drop in κ cr -syn , towards1, is due to the threshold in pair formation – for thoseparticle energies CR photons have too low an energy to air multiplicity in young pulsars κ max , given by eq. (1) in § ζ cr -syn = κ cr -syn κ max (24)can be considered as the efficiency of splitting the en-ergy of the primary particle into pairs. In Fig. 10 weplot ζ cr -syn for the same values of parameters as κ cr -syn in Fig. 9. Despite lower multiplicity for smaller valuesof B , the cascade efficiency is higher, i.e. more of theinitial energy of the primary particle goes into pair for-mation as opposed to the energy of escaping photons andkinetic energy of pairs and primary particles. This trendis discussed in § B (cid:46) few × G the cascade can be quite efficient insplitting a noticeable fraction of primary particle energyinto pairs. For magnetic field ∼ G the fraction ofthe primary particle’s energy going into pair productionsaturates at ∼ ζ cr -syn on (cid:15) ± is similar to the depen-dence of κ cr -syn on (cid:15) ± . PARTICLE ACCELERATION6.1.
Overview of particle acceleration regimes
In this section we will get an estimate for the energyof primary cascade particles. In the following discus-sion we will rely on results of self-consistent modeling ofpair cascades by Timokhin (2010) [T10] and Timokhin& Arons (2013) [TA13]. First, we give a brief overviewof how particle acceleration proceeds according to thesesimulations.Whether and how the pair formation along given mag-netic field lines occurs depends on the ratio j m /j gj of thecurrent density required to support the twist of magneticfield lines in the pulsar magnetosphere (e.g. Timokhin2006; Bai & Spitkovsky 2010), j m ≡ ( c/ π ) |∇× B | , to thelocal GJ current density, j gj ≡ η gj c , where η gj = B/P c is the GJ charge density.For the Ruderman & Sutherland (1975) cascade model,where particles cannot be extracted from the NS sur-face, effective particle acceleration and pair formationis possible for almost all values of j m /j gj (T10). In thespace charge limited flow regime, first discussed by Arons& Scharlemann (1979), pair formation is not possible if0 < j m /j gj <
1, but is possible for all other values of j m /j gj (TA13). Pair formation is always non-stationary:an active phase when particles are accelerated to ultra-relativistic energies and give rise to electron-positron cas-cades – with a burst of pair formation – is followed by a quiet phase when recently generated dense electron-positron plasma screens the electric field everywhere.As pair plasma leaves the active region, it flows intothe magnetosphere, and later into the pulsar wind. Whenthe density of the pair plasma drops below the minimumdensity necessary for screening of the electric field a gapappears – a charge starved region where the electric fieldis very strong, of the order of the vacuum electric field.To screen the electric field, the plasma density must behigh enough to provide both the GJ charge density η gj and the imposed current density j m . The transition be-tween the region(s) still filled with plasma (hereafter wecall it “plasma tail”) and the gap is sharp – plasma stillcapable of screening the electric field moves in bulk, itsdensity is close to the critical density. The plasma den-sity drops abruptly at the gap boundary; within the gapthe particle number density is much smaller than thecritical density (cf. Fig. 22, 23 in § j m /j gj . However, despite these differences, the way inwhich the highest energy particles are accelerated is verysimilar in any regime which allows pair formation stud-ied in T10, TA13. Namely, the size of the charge starvedregion grows as the tail of pair plasma moves, the bulkvelocity of the tail v sets the rate of the gap expansion.Particles entering the gap from the tail are acceleratedin a larger and larger gap, until they are able to producepair-producing photons. The place where these photonsare absorbed and produce pairs is the other boundary ofthe gap. 6.2. Energy of primary particles
In this section we obtain a quantitative estimate forthe maximum energy of accelerated particles using asan example the case of the Ruderman-Sutherland (RS)cascade, when particles cannot be supplied from the sur-face of the NS. As we mentioned before, gap formationand particle acceleration in the space charge limited flowregime, when pair creation is allowed, is very similar tothe RS case and estimates for particle energies obtainedin this section are applicable for the space charge limitedflow with j m /j gj > j m /j gj <
0. The GJ chargedensity is positive and we consider the case when the ra-tio j m /j gj >
0. In Fig. 11 we show a schematic picture ofhow particles are accelerated in this cascade. On the topof each figure we show the electric field in the accelerat-ing region and on the bottom a schematic representationof plasma motion in and around the gap; plot (a) corre-sponds to the time when the electric field screening hasjust started, plot (b) shows a well-developed gap movinginto the magnetosphere. These schematic plots illustrateresults of actual simulations of RS cascades shown in2
Timokhin & Harding (a)
III (b)
IIIaIIb
Figure 11.
Schematic representation of gap formation and evolution for cascades in Ruderman-Sutherland regime with j m /j gj >
0. Seetext for explanation.
Fig. 3,4,11 and 13 in T10.At the beginning of the burst of pair formation, thegap appears at the NS surface and its upper boundary isthe “tail” of plasma left from the previous burst of pairformation, where the particle number density is still highenough to screen the electric field (region I in Fig. 11).Electrons and positrons in this tail are trapped in elec-trostatic oscillations and the bulk velocity of this tail v is sub-relativistic, but for large current densities (aroundor greater than j gj ) it is quite close to c . Electrons fromthis tail which get to the gap boundary are pulled intothe gap and are accelerated toward the NS. As the tailmoves, the gap grows; the current and charge density inthe gap is due to the flux of electrons from the tail andso it remains constant within the gap. The gap growthis stopped when electrons reach an energy high enoughto produce pair-creating photons. This first-generationof pairs start screening the electric field – electrons movetoward the NS and positrons are accelerated toward themagnetosphere and start producing pair creating pho-tons as well (region II in Fig. 11(a)). In numerical sim-ulations (T10) the first-generation positrons moving to-ward the magnetosphere have approximately the sameenergies as the primary electrons which initiated the dis-charge. Because those positrons are ultrarelativistic theypractically co-move with the photons and so new pairsare injected close to their parent particles making a blobof pair plasma moving into the magnetosphere (regionIIa in Fig. 11(b)) . This blob is the lower boundary ofthe accelerating gap, and the gap exists until this blobcatches up with the tail from the previous pair formationcycle. For large current densities this can take a while as v is close to c . Plasma leaking from the blob forms thenew tail (region IIb on Fig. 11(b)).In the discharge described above primary particles aremoving in both directions and initiate cascades towardthe NS (electrons) and the magnetosphere (positrons).As the discharges happen close to the NS surface, thecascade can fully develop only in the direction of themagnetosphere – particles moving toward NS slam ontothe star’s surface before they can produce a lot of pairs.For RS discharges the primary, generation 0, particlesinitiating the full cascade in Fig. 1 are positrons in region see also Fig. 22 in §
10 where we show a snapshot from numer-ical simulations of the cascade corresponding to the stage shownin Fig. 11(b)
IIa in Fig. 11(b). As we mentioned above, the energy ofthose positrons is very close to the energy of the primaryelectrons and here, for the sake of simplicity, we provideestimates only for the energy of primary electrons .The evolution of the electric field in any given point x and moment of time t is given by (see e.g. eq. 1 in TA13) ∂E∂t ( x, t ) = − π ( j ( x, t ) − j m ) ≡ − π ˜ , (25) j is the actual current density along a given magneticfield line and j m is the current density imposed by themagnetosphere. The difference ˜ ≡ j − j m in the gapremains constant. When the upper boundary of the gapmoves with the constant speed v this equation can beintegrated to get the electric field in the gap E ( x, t ) = E ( x , t ) + 4 π ˜ x − xv + 4 π ˜ ( t − t ) (26)Where E ( x , t ) is the electric field within the gap at themoment t at the point x . If we assume that the gapboundary is at x at the moment t , then E ( x , t ) = 0.Electrons enter the gap from above and are quicklyaccelerated by the strong electric field and move withrelativistic speed practically from the moment they leavethe plasma tail. If a particle enters the gap at t (in point x ) its coordinate is x ± = x − c ( t − t ), substituting x ± into eq. (26) the electric field seen by that particle isgiven by E ( x p ) = 4 πc ˜ (cid:16) cv (cid:17) ( x − x ± ) ≡ πη gj ξ j l ± (27)In the last step in eq. (27) we denote with l ± ≡ | x ± − x | the distance traveled by the particle in the gap andintroduce ξ j defined as ξ j ≡ ˜ j gj (cid:16) cv (cid:17) , (28) The reason for both kinds of primary particles, electrons andpositrons, acquiring almost the same energies is that the potentialdrop experienced by each of them is regulated by the process ofpair formation, rather than by the details of their acceleration. Wedid analytical estimates for the final energies of the first-generationpositron based on the model presented in this section; the differ-ence between energies of the primary electrons and first-generationpositrons in the frame of the model is about 2%. air multiplicity in young pulsars j gj is the GJ current density in an aligned rotator j gj ≡ η gj c = BP . (29) ξ j is a factor which shows how stronger/weaker the elec-tric field in the gap is compared to the situation of a staticvacuum gap in an (anti-)aligned rotator, like the one con-sidered by Ruderman & Sutherland (1975). v (cid:39) const isa good approximation to the numerical results and so ξ j (cid:39) const. In cascades along magnetic field lines where j m is close to the local value of j gj in an aligned rotator ξ j ∼
2, for the same situation in a pulsar with inclinationangle of 60 ◦ , ξ j ∼ B (cid:38) G(see Appendix B). If radiation losses are negligible, theparticle’s equation of motion is dpdt = − eE , (30)where p = mcγ is particle’s momentum. In terms of thedistance traveled by the particle in the gap l ± = c ( t − t )with E given by eq. (27), particle equation of motion canbe written as dpdl ± = 4 πec η gj ξ j l ± . (31)Integrating this equation and expressing j gj through pul-sar parameters we get for particle energy (cid:15) ± = 2 πB q λ c c ξ j BP l ± (32)The distance the primary particle travels in the regionof unscreened electric field l gap – the size of the gap asseen by the moving particle – is the sum of the distancethe particle travels before emitting pair-producing pho-tons terminating the gap l e ± , gap and the distance theseprotons travel until the absorption point l γ, gap l gap = l e ± , gap + l γ, gap . (33)For any given particle, the larger the distance l e ± the par-ticle travels to the emission point, the higher the particleenergy and the energy of CR photons it emits, and sothe smaller is the distance traveled by the photon untilthe absorption point l γ . The distance the particle trav-els in the gap l gap is the minimum value of l = l e ± + l γ because once the first pairs are injected the avalanche ofpair creation will lead to screening of the electric field.The photon mean free path l γ can be estimated fromeq. (3) as l γ = 2 χ a ρ c b(cid:15) γ . (34)Photon energy (cid:15) γ depends on the particle energy (cid:15) ± which depends on l e ± according to eq. (32), and so l γ is a function of l e ± . l gap can be found by minimizing l = l e ± + l γ using l e ± as an independent variable: l gap isthe value of l which satisfies dl/dl e ± = 0, l γ, gap and l e ± , gap are the values of l γ and l e ± where l reaches its minimalvalue. Using eq. (33) we can write an equation for l γ, gap dl γ, gap dl e ± , gap = − . (35) If the photon energy depends on l e ± as (cid:15) γ ∝ ( l e ± ) α , theneq. (35) is reduced to l e ± , gap = αl γ, gap , (36)where l γ, gap is expressed in terms of l e ± , gap using eq. (34). l gap is then given by l gap = α + 1 α l e ± , gap . (37)The final energy of the primary particle is given byeq. (32) with l ± = l gap . Please note that because thegap moves, the actual size of the gap (see Fig. 11(a)) is h gap = (cid:16) vc (cid:17) l gap . (38)The energy of the CR photons depends on the particleenergy as (cid:15) ± (eq. 21), the particle energy depends on l ± as l ± (eq. 32), hence, (cid:15) γ ∝ ( l e ± ) and α = 6. Substitutingexpression for (cid:15) ± (eq. 32) into the expression for CR pho-ton energy (cid:15) γ (eq. 21), the latter into eq. (34), and theresulting expression for l γ into eq. (36) with α = 6 afteralgebraic transformations we get the following expressionfor l e ± , gap l e ± , gap = (cid:32) B q λ c c π (cid:33) / χ / a ξ − / j ρ / P / B − / . (39)The size of the gap as seen by the moving particle ac-cording to eq. (37) is l gap (cid:39) × χ / a ξ − / j ρ / P / B − / cm , (40)where B ≡ B/ G and ρ c, 7 ≡ ρ c / cm. Substitut-ing l ± = l gap into eq. (32) we get for the final energy ofparticles accelerated in the gap (cid:15) ± , acc = 4918 (cid:18) πB q λ c c (cid:19) / χ / a ξ / j ρ / P − / B − / (cid:39) × χ / a ξ / j ρ / P − / B − / . (41)The dependence of the primary particles energy on pul-sar period P , filling factor ξ j and the strength of themagnetic field B is very weak, the only substantial de-pendence is on the radius of curvature of magnetic fieldlines. The reason for this is the strong dependence ofCR photon energy on the energy of emitting particles (cid:15) γ ∝ (cid:15) ± , eq. (21). Changes in the threshold energy ofpair producing photons which can stop the gap growthcause only modest variation of the energy of primaryparticles, which explicitly depends on P , ξ j and B , butnot on ρ c , eq. (32). The energy of pair producing pho-tons sets the energy of accelerated particles, and in pul-sars with a strong accelerating electric field the gap willbe smaller than in pulsars with a weaker accelerating Our expression for the energy of primary particles has thesame dependence on ρ c , P and B as the expression for the poten-tial drop in the gap derived by Ruderman & Sutherland (1975),their eq. (23). This is to be expected as in both cases particlesare accelerated by the electric field which grows linearly with thedistance and the size of the gap is regulated by absorption on cur-vature photons in magnetic field. The difference is in the presenceof factor ξ j and a different numerical factor. Timokhin & Harding log ρ c l og B log ϵ ± , acc Figure 12.
Primary particle energy: contours of log (cid:15) ± , acc as afunction of logarithms of the radius of curvature of magnetic fieldlines ρ c in cm and magnetic field strength B in Gauss. We usedthe following values for gap parameters P = 33 ms, ξ j = 2 and χ a = 1 / field. In Fig. 12 we plot the energy of particles (cid:15) ± , acc accelerated in the gap of a pulsar with P = 33 ms asa function of the radius of curvature of magnetic fieldlines ρ c and magnetic field strength B , assuming ξ j = 2and χ a = 1 /
7. The value χ a ≈ / χ a of CR photons emitted by relativistic particles with (cid:15) ± = 2 . × in a magnetic field B = 10 G with ρ c = 10 cm, this is a good estimate for χ a in eq. (41)for young pulsars as the dependence on χ a is very weak.This plot clearly illustrates the dependence of (cid:15) ± , acc on B and ρ c – weaker magnetic field and/or larger radius ofcurvature requires larger photon energies for terminatinggap growth, and so the energy of the primary particlesis larger.We derived eq. (41) under the following assumptions:(i) particles are accelerated freely, i.e. radiation reac-tion can be neglected, (ii) the length of the gap is muchsmaller that the polar cap radius, so that a one dimen-sional approximation can be used, (iii) the magnetic fieldis B (cid:46) . B q ≈ . × G so that the opacity to γB pair creation is described by eq. (2). Constraints on thepulsar parameters (i) and (ii) are derived in Appendix Band Appendix C correspondingly. Plotted on the P ˙ P diagram, Fig. 13, these restrictions select the range ofpulsar period and period derivatives – shown as yellowregion – where all three assumptions are valid. In thisfigure, the one-dimensional approximation (ii) is valid tothe left of the solid line, given by eq. (C5), the approxi-mation (i) of free acceleration above the dot-dashed line,given by eq. (B2), and pulsars with B < . B q are belowthe dotted line. We see that most of young normal pul-sars, including gamma-ray pulsars from the Fermi secondpulsar catalog, fall in this range. − − − − − − − − − ˙ P P
Figure 13. P ˙ P γ -ray pulsars from the second Fermi catalog (Abdoet al. 2013) by red dots. CASCADE MULTIPLICITY PER PRIMARYPARTICLE: SEMI-ANALYTICAL MODELWe now combine the results of § § P , ξ j ,strength B , and ρ c . The multiplicity of CR-synchrotroncascades depends on the energy of the primary particle (cid:15) ± , magnetic field B , and radius of curvature of magneticfield lines ρ c . The only significant dependence of the en-ergy of accelerated particles (cid:15) ± in young pulsars is on theradius of curvature of magnetic field lines ρ c ; the depen-dence on P , ξ j , and B is very weak, see eq. (41). There-fore, when particle acceleration is taken into account, theoverall cascade multiplicity κ can depend substantiallyonly on B and ρ c .In CR-synchrotron cascade changes in (cid:15) ± and ρ c change κ in opposite directions – for higher (cid:15) ± multi-plicity is higher, for larger ρ c multiplicity is lower. Butthe energy of the primary particles accelerated in polarcaps of young pulsars (cid:15) ± is higher for larger radii of cur-vature of magnetic field lines ρ c . Hence, increasing ρ c lowers the cascade multiplicity for a fixed (cid:15) ± , but at thesame time increases (cid:15) ± , which partially compensates thedecrease of cascade multiplicity. Therefore, the final cas-cade multiplicity should have a rather weak dependenceon ρ c , which leaves the magnetic field strength B theonly parameter significantly affecting the multiplicity ofstrong cascades in polar caps of young pulsars.In Fig. 14 we show the final multiplicity as a func-tion of magnetic field strength and radius of curvatureof magnetic field lines. We present plots for three setsof parameters P and ξ j which differ by ∼ an order ofmagnitude. As expected, both pulsar period as well as air multiplicity in young pulsars log ρ c l og B P =
33 ms, ξ = log ρ c P =
33 ms, ξ = log ρ c P =
330 ms, ξ = Figure 14.
Multiplicity of polar cap cascades: contours of log κ as a function of logarithms of curvature of magnetic field lines ρ c in cmand magnetic field strength B in Gauss for three sets of the gap parameters ( P [ms] , ξ j ): (33 , , . , χ a = 1 / the filling parameter ξ j have a very small effect on thefinal multiplicity. For the range of ρ c and B in Fig. 14,the CR-synchrotron cascade has the highest efficiency –primary particles lose most of their energy as CR pho-tons within the distance s cr = R ns where the cascade ispossible – and higher multiplicities cannot be achieved.According to Fig. 14 the cascade multiplicity scales with ρ c roughly as κ ∝ ( ρ c ) ν , with ν (cid:46) /
2. The interval[10 , ]cm represents the most reasonable values of ρ c for any global magnetic field configuration in the pulsarpolar cap. For very different magnetic field configura-tions – a highly multipolar field with ρ c ∼ cm vs. adipole field with ρ c ∼ cm – the multiplicity differsby less than an order of magnitude. The dependence of κ on the magnetic field is stronger, with the maximumreached near B ∼ G.It is a remarkable fact that the multiplicity of the mostefficient cascades is sensitive mostly to the strength of themagnetic field. The multiplicity is not very sensitive to ρ c and for typical pulsar magnetic field of 10 G is around10 . For fixed B the total pair yield – the total numberof particles injected into the magnetosphere – dependsthen only on the flux of primary particles. This is truefor pulsars where the accelerating potential is regulatedby pair production and where cascade operate in CR-synchrotron regime have high efficiency. CASCADE MULTIPLICITY PER PRIMARYPARTICLE: NUMERICAL SIMULATIONSIn §§ P = 33 ms) and magnetic field strength B andradius of curvature of magnetic field lines ρ c having val-ues resulting in high multiplicity. Our goal was to showthat the main assumptions and conclusions of our anal-ysis of polar cap cascades are realistic. More extensive self-consistent numerical studies of the polar cap cascadeswill be done in subsequent papers.As outlined in § If the primary parti-cle energies are known, the full cascade can be modeledusing traditional Monte-Carlo techniques (Daugherty &Harding 1982); to obtain the energies of primary particlesinitiating the CR-synchrotron cascade a self-consistentmodel of the cascade (Timokhin 2010; Timokhin & Arons2013) is necessary.We have performed numerical simulations of time-dependent polar cap pair cascades in a two-step process.In the first step, we use a hybrid Particle-in-Cell/MonteCarlo (PIC/MC) code PAMINA ( P IC A nd M onte-Carlocode for cascades IN A strophysics) to simulate the initialself-consistent electric field generation, particle accelera-tion and electric field screening near the NS to obtainthe distribution functions of the electrons and positronsin the acceleration/screening region. This code includesonly CR of the particles and first generation of pairsneeded to screen the gap, and so does not follow thefull synchrotron cascade. The details of this code are de-scribed in Timokhin (2010); Timokhin & Arons (2013).In the second step, we use another code to simulate thefull pair cascade in the pulsar dipole field above the PC,including both CR of primary particles and synchrotronradiation of pairs. This code, based on the calculation de-scribed in detail in Harding & Muslimov (2011) [HM11],is a Monte-Carlo simulation of the electron-positron paircascade generated above a PC by accelerated particlesin the region of screened electric field. Although HM11included a steady particle acceleration component, thiscomponent is not used in the present calculation. Wetherefore assume that the particles and the further pairsthey create do not undergo any acceleration. This codetakes as input the distribution functions of acceleratedparticles output by the time-dependent PAMINA codethat are moving away from the NS surface and simu-6
Timokhin & Harding lates the combined cascade from all of the particles. Al-though our setup is capable of calculating full cascadesgenerated by primary particles with arbitrary distribu-tion functions, in the simulations described in this sectionwe used a monochromatic injection of primary particles.On the one hand, the energy distribution of the most en-ergetic primary particles which produce the bulk of thepairs in many cases is close to monochromatic (see e.g. §
10, Fig. 23), and on the other hand this enables us tocompare numerical simulations with predictions of oursemi-analytical theory. The energy of the primaries wascalculated from the self-consistent model however.The MC code first follows the primary particle in dis-crete steps along the magnetic field line at magnetic co-latitude θ , starting from the location x and particle en-ergy (cid:15) ± at time t peak at the peak of the pair productioncycle in PAMINA code, computing its curvature radia-tion. The steps ∆ x are set to the minimum of a fraction0 . n cr , is determined bythe energy loss rate and average energy in that bin. Thepairs produced by the photon, or the escaping photonnumber, is then weighted by n cr . The created pair isassumed to have the same direction and half the energyof the parent photon. Although the CR photons are ra-diated parallel to the magnetic field, they must acquirea finite angle to the field before producing a pair, so thecreated pairs have finite pitch angles at birth. Each mem-ber of the pair emits a sequence of cyclotron and/or syn-chrotron photons, starting from its initial Landau stateuntil it reaches the ground state, assuming the positionof the particle remains fixed (given the very rapid radi-ation rate). As described in HM11, when the pair Lan-dau state is larger than 20, the asymptotic form of thequantum synchrotron rate (Sokolov & Ternov 1968) isused to determine the photon emission energy and finalLandau state. When the Landau state is below 20, thefull QED cyclotron transition rate (Harding & Preece1987) is used. At large distances above the NS surface,when the magnetic field drops below 0 . B q , we short-cut the individual emission sequence and use an expres-sion for the spectrum of synchrotron emission for an elec-tron that loses all of its perpendicular energy (Tademaru1973). Each emitted photon is then propagated throughthe magnetic field from its emission point until it pairproduces or escapes. The next generation of pairs arethen followed through their synchrotron/cyclotron emis-sion sequence. By use of a recursive routine that is calledupon the emission of each photon, we can follow an arbi-trary number of pair generations. The cascade continuesuntil all photons from each branch have escaped. As eachmember of each created pair completes its synchrotronemission, its ground-state energy, position and genera-tion number are stored in a pair table. As each photoneither pair produces or escapes, its energy, generationand position of pair creation or escape are stored in ta-bles for absorbed and escaping photons. The photonsand pairs from all accelerated particles are summed to- gether to produce the complete cascade portrait at thattime step.The NS magnetic field in the MC code described aboveis a distorted dipole with an azimuthal ( φ ) componentwhich is off-set from the center of the NS. The magneticfield is given by B = B (cid:18) R ns r (cid:19) × (cid:20) e r cos[ θ (1 + a )] + e θ
12 sin[ θ (1 + a )] − e φ ε ( θ + sin θ cos θ ) sin( φ − φ ) (cid:21) , (42)where B is the surface magnetic field strength at themagnetic pole, r is the radial coordinate, a = ε cos( φ − φ ) is the parameter characterizing the distortion of po-lar field lines, and φ is the magnetic azimuthal angledefining the meridional plane of the offset PC. The pa-rameter ε sets the magnitude and the parameter φ setsthe direction of asymmetry of this azimuthal component.Setting ε = 0 gives a pure dipole field structure, while anon-zero value of ε produces an effective offset of the PCfrom the dipole axis in the direction specified by φ . Fornon-zero ε , the radius of curvature of the magnetic fieldlines is smaller than dipole in the direction of the offsetand larger than dipole in the direction opposite to thedirection of offset. This particular parametric form forthe magnetic field was used in simulations of stationarycascades in HM11, it was chosen to account for distortionof the shape of the PC caused by currents flowing in pul-sar magnetosphere. Such azimuthal asymmetries in thenear-surface magnetic field is caused by the sweepback ofthe field lines near the light cylinder due to retardation(e.g. Dyks & Harding 2004) and currents (e.g. Timokhin2006; Bai & Spitkovsky 2010; Kalapotharakos et al. 2014)or, additionally, by asymmetric currents in the NS.We did not perform a systematic study of all parame-ter space with our numerical simulations, which will bedone elsewhere, with our numerical simulations we testassumptions and predictions of our semi-analytic cascademodel. Any form of magnetic field with adjustable radiusof curvature of magnetic field lines would serve our pur-poses, but using the magnetic field given by eq. (42) al-lows comparison with the most recent simulations in theframe of the previous-generation cascade models HM11.We explored cases of pair cascades both for pure dipolefields and for azimuthally distorted fields. Multiplicitiesobtained from the numerical simulations agree reason-ably well with the semi-analytic model, within a factorof a few. As an example we describe in detail results ofparticular simulations with pulsar parameters yieldinghigh multiplicity for a cascade at the peak of the paircreation cycle. The magnetic field is B = 10 G and ismoderately distorted, with the offset ε = 0 . ρ c = 8 . × cm. The initial energy of primaryparticles from PAMINA simulations is (cid:15) ± = 2 . × .In Figures 15 and 16 we show a set of five plots whichwe call “cascade portraits”. These plots illustrate dif-ferent aspects of the cascade development by showingmoments of the photon or particle distribution function f ( x, E ). The top panel shows the number of particles air multiplicity in young pulsars − − − − x [ R NS ] E [ m e c ] dN ( x , E ) e ± E [ m e c ] x [ R NS ] dN ( x ) N ( ≤ x ) E f ( x i , E ) E f ( E ) all gengen 0gen 1gen 2gen 3gen 4 x = . x = . x = . x = . Figure 15.
Cascade portrait of electron-positron pairs.
Top panel : number of particles produced in each energy and distance bin dN ( x, E )color coded in logarithmic scale according to the colorbar on the right. Middle left : E f ( x i , E ) – energy distribution of particles producedat four distances x i , i = 1 . . .
4; different line styles correspond to different distances according to the left plot legend.
Bottom left : E f ( E )– energy distribution of all particles produced in the cascade. Middle right : dN ( x ) – differential pair production rate – number of particlesproduced in a distance bin. Bottom right : N ( ≤ x ) – total number of particles produced up to the distance x . Color lines in plots for E f ( E ), dN ( x ), and N ( ≤ x ) show contributions of different cascade generations, lines are color-coded according to the right plot legend.Thick black lines show contributions of all cascade generations. x is the distance from the NS normalized to NS radius R ns and E is particleenergy normalized to m e c . Particle number density is normalized to n gj . Parameters of this simulation: pulsar period P = 33 ms; themagnetic field in the PC has B = 10 G and ρ c = 8 . × cm; initial energy of primary particles (cid:15) ± = 2 . × . produced in each energy and distance bin dN ( x, E ) = f ( x, E ) dx dE, (43)as a 2D color map. The number of particles is colorcoded in logarithmic scale according to the color bar onthe right. The middle left plot shows the energy distri-bution E f ( x i , E ) of particles produced at four distances x i , i = 1 . . .
4; different line styles correspond to differentdistances according to the left plot legend. These spec- tra are essentially cross-sections of the map of dN ( x, E )(multiplied by particle energy) along four lines shown inthe plot for dN ( x, E ). The bottom left plot shows the en-ergy distribution of all particles produced in the cascade( dN ( x, E ) integrated along x direction and multiplied by E ) E f ( E ) = E (cid:90) x max d ˜ x f (˜ x, E ) . (44)8 Timokhin & Harding − − − − − − − − − − − − − x [ R NS ] E [ m e c ] dN ( x , E ) γ esc E [ m e c ] x [ R NS ] dN ( x ) N ( ≤ x ) E f ( x i , E ) E f ( E ) all gengen 0gen 1gen 2gen 3gen 4gen 5 x = . x = . x = . x = . Figure 16.
Cascade portrait of escaping photons for the same cascade as in Fig. 15. Notations and normalizations are the same as inFig. 15.
In this and the following plots, colored lines show contri-butions of different cascade generations; lines are color-coded according to the right plot legend. The middleright plot shows the differential pair production rate –number of particles produced in a distance bin ( dN ( x, E )integrated along the E direction) dN ( x ) = dx (cid:90) E max f ( x, E ) dE . (45)The bottom right plot shows the cumulative pair produc-tion rate – total number of particles produced up to thedistance xN ( ≤ x ) = (cid:90) x d ˜ x (cid:90) E max f (˜ x, E ) dE . (46) Figure 15 shows the cascade portrait of the pairs (we donot differentiate between electrons and positrons). Thecascade extends to about ∼ R ns , where it dies out com-pletely as the magnetic field strength and the acceler-ated particle energy decrease. The pair number growsvery quickly, within a few tenths of a NS radius, andthen saturates at κ ≈ . × . The majority of pairsare produced at distances < R ns (see plot for N ( ≤ x )),which supports the assumption about the length of thecascade zone being ∼ R ns made in § (cid:46) R ns , up to six in thiscase . The occurrence of most of the cascade generations There are too few pairs produced in the 6th generation to showin the plot; curves corresponding to this generation are below the air multiplicity in young pulsars ∼ R ns is the evidence that the photonmfp in a strong cascade is indeed small, l γ (cid:28) R ns , so thatthe cascade initiated by any given particle goes throughseveral generations within the distance ∼ R ns . The largeextent of the cascade zone relative to l γ is due to con-tinuous injection of pair-producing CR photons (see theblue line in the plot of dN ( x ); generation 0 pairs areproduced by CR photons). The largest contribution topair multiplicity in this case comes from generation 1,pairs created by the first synchrotron photons (see plotsfor dN ( x ) and N ( ≤ x )). In our simulations for differentpulsar parameters, contributions of generation 1 and 2to the pair multiplicity sometimes become comparable;for weaker cascades the relative contribution of genera-tion 0 is higher than in this case, however we did notsee generations 3 and higher producing the majority ofpairs. The number of cascade generations is not verylarge in any of our simulations (several at most), but ineach generations high numbers of pair producing photonsare emitted which results in high multiplicity.The energy of created pairs decrease with distance (seeplots for dN ( x, E ) and E f ( x i , E )), mostly because ofenergy losses of primary particles which results in lowerenergy CR photons. The pair spectrum extends downto a few mc , since the cascade is very efficient at con-verting initial pair energy into more photons (and pairs).Degradation of pair energies through cascade generationsdiscussed in § E f ( E )– the maximum pair energy systematically decreases withcascade generations at all distances.Figure 16 show the portrait of photons escaping thecascade. Photon generation 0 is CR while the highergenerations ( ≤
1) are synchrotron/cyclotron radiation.Although the highest energy photons are produced near-est the NS surface, these photons are absorbed by pairproduction attenuation so that the spectra at the lowestaltitudes show sharp cutoffs near 100 mc . This cutoff isclearly visible in the plot for Ef ( x i , E ) for x = 0 . R ns shown by a dashed line. In the plot for dN ( x, E ) thecutoff is evident as a sharp horizontal boundary of thecolored region for x (cid:46) . R ns . The escaping photon en-ergies increase with distance from the NS surface, as themagnetosphere becomes more transparent. The highestescaping photon energies are produced near the end ofthe cascade, at around 4 NS radii. The highest energyphotons escaping the cascade are CR photons (blue linein the plot for Ef ( E )). Synchrotron radiation is emittedby pairs right after their creation, so that the pair forma-tion and synchrotron radiation end at the same distance– in the plot for dN ( x ), the number of photons in eachgeneration drops at large distances in accordance withthe drop of number of injected pairs shown on a similarplot in Fig. 15. Above ∼ R ns the emitted CR photonenergies drop as the primary particles continue to loseenergy. The spectrum of escaping photons also broadensat the lower end because, as the magnetic field decreases,so does the cyclotron energy which sets the lower limit ofthe synchrotron spectrum. Thus the lowest and the high-est energy escaping photons are produced at the largestdistances from then NS, cf. spectra at different x i in theplot for Ef ( x i , E ). The bulk of high energy emission lower limit of all plots in Fig. 15; this generation shows in theportrait for photons, Fig. 16
44 55 66 8812 12 log ϵ ± l og B log λ RICS
Figure 17.
Distance (in cm) over which a particle loses its energyvia resonant inverse Compton scattering. Contours of log λ RICS are plotted as a function of logarithms of particle energy (cid:15) ± andmagnetic field B in Gauss for a NS surface temperature T = 10 Kand µ s = 0 . from the PC cascade comes from synchrotron radiationof pairs in generations 1,2, and 3.Quantitatively, our semi-analytic theory compareswith this particular numerical simulation as follows. Forpulsar parameters used in this simulation, the energyof primary particles according to eq. (41) should be (cid:15) , a ± ≈ . × , which is ≈ . (cid:15) , num ± ≈ . × . For the multiplicity ofthe CR-synchrotron cascade started by primary particleswith monochromatic energies (cid:15) , num ± , the semi-analyticmodel gives κ a cr -syn ≈ . × , from eq. (23), whichis also about 2 times higher than the value obtainedin numerical simulations κ num ≈ . × . The com-bined model from §
7, which uses the analytic modelfor particle acceleration as an input for the semi-analyticmodel of CR-synchrotron cascade, predicts for the mul-tiplicity κ a ≈ . × , see Fig. 14, which is ≈ B and ρ c (which is derived from B ) depend on the distanceaccording to eq. (42), photon absorption decreases andbecomes less efficient with the distance. We think thatfor such a simple model the agreement with the numeri-cal simulations is reasonable and the model can be usedfor estimates of multiplicities in young energetic pulsars. RESONANT INVERSE COMPTON SCATTERINGAnother emission mechanism for relativistic particlesbesides curvature and synchrotron radiation is inverse0
Timokhin & Harding e ± γ CR e ± γ syn γ RICS e ± e ± γ syn γ RICS γ syn γ RICS . . . . . . . . . . . . . . .
Figure 18.
Diagram showing the general chain of physical pro-cesses in a strong polar cap cascade. Cascade generations are shownon the left – numbers connected by double arrows. In each genera-tion particles e ± (electrons and/or positrons) produce photons ( γ cr – via curvature radiation, γ syn – via synchrotron radiation, γ RICS – via Resonant inverse Compton scattering), which are turned intopairs of the next cascade generation. The CR-synchrotron cascadesstudied in detail in this paper are shown by solid arrow, dashed ar-rows show RICS initiated branches which are discussed only in § Compton scattering (ICS). In strong magnetic fields typ-ical for pulsar polar caps, ICS can occur in the resonantregime, when the photon energy in the electron’s restframe is equal to the cyclotron energy. The cross-sectionfor scattering of such photons is greatly enhanced com-pared to that of non-magnetic scattering. It has beennoted that resonant ICS (RICS) with the soft thermalphotons from the NS surface is important for high-energyemission from pulsar polar caps (Sturner 1995; Zhang &Harding 2000) and can be important in the developmentof polar cap cascades because for quite a wide range ofpulsar parameters, scattered photons can be above pair-formation threshold (e.g. Sturner et al. 1995; Zhang &Harding 2000). In this section we argue that althoughRICS can play a role in the development of polar cap cas-cades, it never becomes the dominant source for pair mul-tiplicity and, therefore, considering only CR-synchrotroncascades provides adequate estimates for pair multiplic-ity in strong cascades of normal pulsars. A detailed studyof the role of RICS in polar cap cascades will be presentedin a subsequent paper.First, let us consider the efficiency of RICS in trans-forming particle kinetic energy into radiation. The dis-tance over which a particle loses most of its energy toRICS is given by (Zhang & Harding 2000; Sturner 1995;Dermer 1990): λ RICS = − . (cid:15) ± T − B − × log ρ l og B log [ ϵ γ ,RICS ( ) / ϵ γ ,esc ] Figure 19.
Ratio of of the characteristic energy of RICS pho-tons (cid:15) (1) γ, RICS emitted by the first generation pairs to the energy ofescaping photons: contours of log (cid:104) (cid:15) (1) γ, RICS /(cid:15) γ, esc (cid:105) are plotted asa function of of logarithms of the radius of curvature of magneticfield lines ρ c in cm and magnetic field strength B in Gauss. Weused the following values for gap parameters P = 33 ms, ξ j = 2and χ a = 1 / ln − (cid:20) − exp (cid:18) − B (cid:15) ± T (1 − µ s ) (cid:19)(cid:21) cm (47)where T is the temperature of the NS surface in unitsof 10 K, B is the magnetic field strength in units of10 G, and µ s = cos θ s , where θ s is the angle betweenthe momenta of the scattering photon and particle in thelab frame. If the NS is young, its surface temperaturecomes mostly from cooling and should be about 10 K. Itemits X-ray photons necessary for RICS from the wholesurface, and for particles at a distance comparable to theNS radius the range of µ s is quite large. For older NS,with full surface temperatures below few × K only thepolar cap region, heated by the backflow of acceleratedparticles up to ∼ few × K (e.g. Harding & Muslimov2001, 2002), can emit enough photons for RICS to be-come important. In the latter case when the particlereaches a distance comparable to the width of the polarcap r pc (cid:39) . × P cm, where P is pulsar period in sec,the range of µ s gets very small and photons quickly getout of resonance. So, for young NSs RICS is an impor-tant radiation process if λ RICS ∼ R ns ; for old NSs thiscondition changes to λ RICS ∼ r pc In Fig. 17 we plot λ RICS as a function of particle en-ergy (cid:15) ± and magnetic field B , for T = 10 K and µ s = 0 . (cid:15) ± > , lose a negligible amount of their energy viaRICS, and curvature radiation is the dominant emissionmechanism for generation 0 of strong cascades. For coldNSs, RICS of photons from heated polar caps might bean important emission mechanism only for a very nar-row energy range of low energetic particles, in one ofthe later cascade generation – the parameter space be- air multiplicity in young pulsars λ RICS = 10 contours is rather small – and formost values of B the scattered photons will be below pairformation threshold. Hence, RICS can be completely ne-glected in strong polar cap cascades of cold NSs. For hotNSs RICS becomes an important emission mechanism fora wide range of moderate particle energies (cid:15) ± < – theenergy range between contours of λ RICS = 10 in Fig. 17is quite wide. In the latter case, the diagram for physicalprocesses in a strong polar cap cascade can have the gen-eral form shown in Fig. 18, with RICS photons in somecases carrying non-negligible energy starting from cas-cade generation 1. The extension of the semi-analyticalanalysis of CR-synchrotron cascade developed in §§ (cid:15) ± in the RICS regime is (e.g. Zhang & Harding 2000) (cid:15) γ, RICS = 2 (cid:15) ± , f b , (48)where (cid:15) ± , f , given by eq. (12), is the kinetic energy ofa particle moving along a magnetic field line – the finalenergy of freshly created pairs after they emit all per-pendicular to B energy via synchrotron radiation. Us-ing eqs. (48), (12) together with eq. (21) for the energyof CR photons and eq. (41) for the energy of the pri-mary particle, we get an upper limit on the energy of thegeneration 1 RICS photons – the highest energy RICSphotons. In Fig. 19 we plot the ratio of the energy ofgeneration 1 RICS photons (cid:15) (1) γ, RICS to the energy of pho-tons escaping from the cascade zone (cid:15) esc as a functionof the radius of curvature of magnetic field lines and thestrength of the magnetic field B . It is easy to see thatfor magnetic fields weaker than a few × G, even thehighest energy RICS photons are not capable of produc-ing electron-positron pairs. For stronger magnetic fields,however, RICS photons do contribute to pair multiplic-ity.The characteristic energy of RICS photons in termsof the energy of the previous generation photon can beobtained by substituting (cid:15) ± , f into eq. (48) (cid:15) ( i +1) γ, RICS = (cid:15) iγ b (cid:20) (cid:16) χ a b (cid:17) (cid:21) − / . (49)This equation describes the energy degradation in eachcascade generation for the RICS process. In Fig. 20 weplot the ratio of the characteristic energies of synchrotronand RICS photons (given by eqs. (49) and (16) corre-spondingly) produced by pairs created by the same par-ent photon as a function of the parent photon energy (cid:15) ( i ) γ and magnetic field strength B . The energy of RICS pho-tons are always smaller than the energy of synchrotronphotons, and for B < G significantly so. Because ofthe much faster energy degradation in the RICS process,cascade branches initiated by RICS photons are in gen-eral shorter than those initiated by synchrotron photons.In later cascade generations RICS photons freely escapethe cascade zone while synchrotron photons emitted by .051 2 34 log ϵ γ ( i ) l og B log [ ϵ γ ,syn ( i + ) / ϵ γ ,RICS ( i + ) ] Figure 20.
Ratio of the characteristic energies of synchrotron (cid:15) ( i +1) γ, syn and RICS (cid:15) ( i +1) γ, RICS photons emitted by freshly cre-ated pairs (( i + 1) th generation cascade photons): contours oflog (cid:104) (cid:15) ( i +1) γ, syn /(cid:15) ( i +1) γ, RICS (cid:105) are plotted as a function of logarithms of theparent photon energy (cid:15) ( i ) γ and magnetic field strength B in Gaussfor ρ c = 10 cm. .5 1 310 3070 log ϵ γ l og B ϵ ⟂ / ϵ || Figure 21.
Ratio of perpendicular to parallel to B energy offreshly created pairs: contours of (cid:15) ⊥ /(cid:15) (cid:107) as a function of logarithmsof the parent photon energy (cid:15) γ and magnetic field strength B inGauss for ρ c = 10 cm. the same particles are absorbed, still splitting the en-ergy of the parent photons into pairs. Therefore, RICScascade branches should be in general less efficient insplitting the energy than synchrotron ones.2 Timokhin & Harding
Finally let us address the question of how much en-ergy is going into RICS branches of the cascade. Theenergy powering RICS branches is the kinetic energyof pairs moving along magnetic field lines, while syn-chrotron branches are powered by the perpendicular en-ergy of freshly created pairs. From eqs. (12), (13) we getfor the ratio of the perpendicular to parallel pair energies (cid:15) ⊥ (cid:15) (cid:107) = (cid:20) (cid:16) χ a b (cid:17) (cid:21) − / − . (50)As χ a > b (see eq. (8)), the minimum value for this ratiois √ − (cid:39) . (cid:15) ⊥ /(cid:15) (cid:107) as a functionof the energy of the parent photon and magnetic fieldstrength B . For most of the parameter space in eachpair creation event, the fraction of energy going into a theRICS cascade branch is smaller than those going into thesynchrotron branch. Even in the case when more energyis left in the pairs’ parallel motion, the energy available tothe RICS branch is only 1 . χ a is close to b , i.e. pair formation occursnear the kinematic threshold and is not very efficient (see § (cid:38) few × G, when pairformation happens close to kinematic threshold and thecascade multiplicities are lower than for weaker magneticfields. The final multiplicity of the total cascades shouldbe less than twice that of the pure CR-synchrotron cas-cade in the best case and so the pure CR-synchrotroncascades studied in this paper provide good estimatesfor the multiplicity of strong polar cap cascades.
FLUX OF PRIMARY PARTICLES AND PAIRYIELDAs we have shown above, for the most efficient polarcap cascades the total pair yield for a given value of themagnetic field B depends mostly on the flux of primaryparticles. According to self-consistent models of polarcap acceleration zones (T10, TA13), particle accelera-tion is intermittent and the pattern of plasma flow andacceleration efficiency depends on the ratio of the currentdensity imposed by the magnetosphere to the GJ currentdensity j m /j gj as well as the boundary conditions at theNS surface – whether particles can be extracted from thesurface or not.There are essentially three qualitatively differentregimes of plasma flow that determine the flux of primarypair-producing particles: (i) in space charge limited flowwith 0 < j m /j gj < (cid:15) ± ∼ few, and no pairs are produced, (ii) inspace charge limited flow with j m /j gj > ∼ j gj /e is accel-erated through the gap while the gap is moving towardthe NS, (iii) in all other cases – i.e. for any current den-sity in Ruderman-Sutherland model and for j m /j gj < bulk of pair-producingparticles IIIaIIb
Figure 22.
Snapshot of the phase space for Ruderman-Sutherlandcascade at the end of a discharge. Cascade an pulsar parameters: j m /j gj = 1 B = 10 G, P = 33 ms, ξ j = 1. Horizontal axis– particle positron x , vertical axis – particle momentum normal-ized to m e c ; the vertical axis is logarithmic except for the regionaround zero momentum ( − < p < p − vs x – electrons, p + vs x – positrons, p γ vs x – photons. Par-ticle number density is color-coded according to the color map onthe right in units of n gj . Particles which produce most of the pairsare positrons inside the area surrounded by dotted line marked as“bulk of pair-producing particles”. On the top of the plots we showthe sketch of the structure of the acceleration zone from Fig. 11(b). air multiplicity in young pulsars bulk of pair-producingparticles Figure 23.
Snapshot of the phase space for cascade in SpaceCharge Limited Flow regime. Cascade an pulsar parameters: j m /j gj = 1 . B = 10 G, P = 33 ms, ξ j = 0 .
25. Types ofplots and notations are the same as in Fig. 22. formation of the gap, and then no significant amount ofprimary particles is created until the formation of thenext gap.In Fig. 22 we plot as an example for case (iii) a phaseportrait and densities of plasma and photons in thephase space for a Ruderman-Sutherland cascade with j m /j gj = 1. The process of gap formation for this caseis described at the beginning of § ∼ j gj /e and their contribution to the pair production can be ne-glected. The blob of primary particles which was createdduring the gap formation has a density of a few n gj in its densest parts and a size comparable to the gap’s height.Those particles – in this case positrons in the area sur-rounded by a dotted line on the plot for positron phasespace density in Fig. 22 – create the vast majority ofpairs. The density of primary particles in the blob ishigh but no new primary particles are created until thenext gap is formed, the duty cycle of the cascade is small.The resulting flux of primary particles averaged over theperiod of gap formation is rather low.In case (ii), in space-charge limited flow with super GJcurrent density, the situation is qualitatively different inthat during the lifetime of the gap, and not only duringthe time the gap is forming, as in case (iii), a significantconstant flux of particles is going through the accelera-tion zone. The duty cycle of such a cascade should besubstantially higher than in any other types of cascades.As an example of such a cascade we plot in Fig. 23 aphase portrait and the densities of plasma and photonsin phase space for a cascade in space-charge limited flowregime with j m /j gj = 1 .
5. The gap is formed at some dis-tance from the NS and moves toward it. When the firstparticles are formed the process of electric field screen-ing proceeds in a similar way to the case of Ruderman-Sutherland cascades; the blob of ultrarelativistic parti-cles is created and particles leaking from it create a tailof mildly relativistic plasma screening the electric fieldbehind the blob; this blog moves toward the NS creat-ing pairs. However a constant flux of particles extractedfrom the NS surface – in this case electrons, visible asa line-like feature in the top two panels of Fig. 23 – areaccelerated in the gap. The density of these electronsis high (for the case shown in Fig. 23 it is (cid:39) . n gj )and these particles, and not the particles from the blob,produce most of the pairs.Because of the intermittency of pair formation, theresulting multiplicity must be adjusted by the relativefraction of time during which primary particles are pro-duced. If τ active is the time of active particle accelerationand T cascade is the time between the beginning of suc-cessive bursts of pair creation, then the pair multiplicityas discussed in previous sections should be multiplied bythe attenuation factor f κ = τ active T cascade (51)to get the average multiplicity of pair cascades.The existing self-consistent simulations of particle ac-celeration (T10, TA13) are inconclusive about the cas-cade repetition rate. The numerical resolution in thesesimulations was inadequate for that purpose – becauseof copious pair formation the Debye length of plasmaat some point became smaller than the cell size and theformation of the plasma tail could not be simulated accu-rately; the repetition rate, however, is determined by themildly relativistic plasma in the plasma tail, as the nextburst of pair formation starts only when plasma leavesthe polar cap region. Future numerical simulations wouldaddress this issue and provide accurate values for cascadeduty cycles; meanwhile here we will try to give a roughestimate for the cascade repetition rate using the follow-ing simple physical arguments.The strongest pair formation can continue up to thedistance of the order of a few NS radii R ns . Althoughthe exact physical mechanism responsible for formation4 Timokhin & Harding of the mildly relativistic plasma tail is not clear (see pre-vious paragraph) it seems to us plausible to expect thatwhen plasma injection due to pair formation stops, rever-sal of some pairs toward the NS – plasma leakage fromthe blob to the tail – should taper off as well. So, thetime when plasma is filling the polar cap region is of theorder of ∼ R ns /c . The mildly relativistic pairs whichfill the polar cap region after the blob of plasma hasmoved away are moving with relativistic velocities, sothe time the plasma needs to clear out is also of the or-der of ∼ R ns /c . Therefore, the cascade repetition rateseems to be T cascade ∼ few × R ns /c For the case (iii), in the flow regime with vacuum gaps– Ruderman-Sutherland cascades or space-charge limitedflow in return current regions – particles are acceleratedonly during formation of the gap, the time of active par-ticle acceleration is quite short. The longitudinal size ofthe blob of primary particles is about the size of the gap h gap , and the primary particle density in the blob is a few n gj . All primary particles in the blob move in the samedirection and the blob passes any given surface normalto the field lines during time τ active ∼ h gap /c . If the nextgap will form after the blob of primary particles moves afew R ns away, the attenuation factor should be f κ ∼ h gap R ns (cid:39) × − χ / a ξ − / j ρ / P / B − / (52)For the Crab pulsar this attenuation factor is around0 .
001 if we assume P = 33 ms, B = 3 × G, χ a =1 /
7, and ξ j = 1 (for inclination angle 60 ◦ and currentdensity j m (cid:39) j gj ). The resulting multiplicity of suchcascades in Crab would be (cid:46) n gj .In the space-charge limited flow regime with super-GJcurrent density, case (ii), the cascade efficiency shouldbe much higher. The gap appears at some distance andmoves toward the NS (see § f κ ∼ / few – much higher than given by eq. (52). So,polar cap regions with super-GJ current density shouldhave a pair yield of about 10 n gj .Let us summarize our findings. Pulsars with polar capcascades operating in the Ruderman-Sutherland regimewould not be efficient pair producer, the highest yieldshould be less that ∼ n gj . If the polar cap cascadesoperate in the space charge limited flow regime, then inregions with super-GJ current densities the pair yieldis quite high, around 10 n gj ; in the regions with returncurrent (anti-GJ current density) the yield should be lessthat ∼ n gj , and in regions with sub-GJ current densi- ties no pair plasma would be produced at all. We there-fore conclude that the maximum pair yield of a youngpulsar would be less that 10 n gj , as only a fraction ofthe polar cap can have super-GJ current density.Pair yield is mostly affected by the duty cycle of thecascade, and not by pulsar parameters such as magneticfield strength B , radius of curvature of magnetic fieldlines ρ c , or pulsar period P . This holds only for theregime in which the gap height is much smaller than thePC radius, true for all young pulsars. When the gapheight approaches and exceeds the PC radius, the pairyield drops quickly, as shown in Figure 9. The distri-bution of j m /j gj in the polar cap is determined by thepulsar inclination angle and we expect that inclinationangle should be the most important factor determiningpair yield of a young pulsar. DISCUSSIONWe have performed a systematic study of electron-positron pair cascades above pulsar polar caps for a vari-ety of input parameters including surface magnetic field,pulsar rotation period, primary particle energy and mag-netic field radius of curvature. We have also studied herefor the first time the multiplicity of self-consistent paircascades, i.e. those that are capable of generating cur-rents consistent with global magnetosphere models.We find that pair multiplicity is maximized for a mag-netic field strength near 10 G, independent of the otherparameters. This value of field strength strikes a balancebetween maximizing the fraction of photons that pairproduce – the weaker the magnetic field the higher thenumber of escaping photons – and the fraction of pho-ton energy going into pair production – in stronger fieldsphotons are absorbed when they have smaller angles to B , and created pairs, having smaller perpendicular to B momenta, emit less synchrotron photons, which producethe next generation of pairs. This is true for curvatureradiation - synchrotron cascades, which should be thedominant source of pairs in young energetic pulsars. Forstronger magnetic fields B (cid:38) × G resonant inverseCompton scattering of soft X-ray photons emitted by theNS surface can tap some of the pair’s energy parallel to B and increase the cascade multiplicity. RICS cascadebranches, however, are less efficient than the synchrotronbranches; this together with the decrease of the photonabsorption cross-section for near threshold pair creationin strong magnetic fields should not change the fact thepolar cap cascade multiplicity reaches its maximum for B ∼ G.We find that the pair multiplicity at the peak of thecascade cycle, κ ∼ , is remarkably insensitive to pul-sar period, magnetic field and radius of curvature of mag-netic field lines. The reason for this is self-regulation ofthe accelerator by pair creation: for pulsar parametersresulting in more efficient pair production, the size ofthe acceleration gap is smaller and the primary particleenergy is lower and vice-versa. The most important fac-tor determining the multiplicity of polar cap cascades isthe flux of primary particles which depends on the “dutycycle” of the particle acceleration in time-dependent cas-cades. Estimating the “duty cycle” we find that the time-averaged pair multiplicity is limited to ∼ for thecase of space charge-limited flow (free particle extractionfrom the NS) with super Goldreich-Julian current density air multiplicity in young pulsars j/j gj >
1) and to only ∼ for the case of Ruderman-Sutherland gaps (no particle extraction from the PC)and space charge-limited flow with anti Goldreich-Juliancurrent density ( j/j gj < ∼ ), that is largelyindependent of their period and magnetic fields, is verydifferent from previous predictions from steady cascadesthat produce a range of multiplicities (10 − ) thatdepend strongly on pulsar parameters. This differenceprimarily results from much smaller gap heights in thetime-dependent cascades, that keep the gap size muchless than the PC radius over a large parameter range.This in turn results in a nearly uniform magnetic fieldstrength throughout the gaps, higher accelerating elec-tric fields, higher primary particle energies and more ef-ficient pair production – primary particles are acceleratedfaster and photons are injected in the region of strongermagnetic field than in previous steady cascade models.We have also numerically simulated the photon spec-tra from self-consistent polar cap pair cascades. Our re-sults show that the cascade photon spectra from the morecompact gaps nearer to the neutron star surface have cut-offs that are around 10 - 100 MeV. This is at the lowerend the Fermi energy band (30 MeV - 300 GeV), whichmay explain why
Fermi has not seen strong evidence for γ -ray emission from the PC. High energy emission frompair cascades is thus expected to occur in lower energybands, below 100 MeV. PC cascade emission may havebeen detected recently from PSR J1813-1246 (Marelliet al. 2014) at X-ray energies by XMM-Newton andChandra. The unusual X-ray light curve that shows twopeaks separated by 0.5 in phase but offset by a quar-ter of a period with the γ -ray peaks can be explainedthrough a geometric model placing the γ -ray emission in the outer magnetosphere and the X-ray emission at loweraltitude above the PCs, and could be coming from thePC cascades.Our results show that PC cascades are more efficientin producing electron-positron pairs that was previouslyassumed. However, even the higher pair multiplicity of10 is not enough to account for the fluxes of parti-cles needed to explain the synchrotron radiation fromPWNe. Based on a multiplicity of 10 , we estimate thatthe Crab pulsar produces a pair flux from each PC ofabout 2 × pairs s − . The flux from both PCs is anorder of magnitude smaller than the pair flux required toaccount for the radiation from the nebula, which is esti-mated to be about ∼ × pairs s − (de Jager et al.1996). Cascades in the outer magnetosphere are not veryefficient pair producers (e.g. Hirotani 2006), and so theinjection of plasma by pulsars can not account for thepopulation of particles in PWNe emitting at radio wave-lengths. These radio emitting particles must then have adifferent origin from particles emitting at shorter wave-length; for example, they might be picked up from thegas filaments in the supernova remnant or be remnantsof some unknown acceleration mechanism in the earlyhistory of the nebula (Atoyan & Aharonian 1996).Another implication of our results concerns observa-tions of cosmic-ray electrons and positrons at Earth thathave shown an excess of positrons over what can beproduced in secondary cosmic-ray interactions (Adrianiet al. 2009; Accardo et al. 2014), indicating the exis-tence of primary positron sources in the Galaxy. Var-ious studies (e.g. Gendelev et al. 2010) have estimatedthat PWNe could account for the excess of cosmic raypositrons. With higher pair multiplicity from young pul-sars, and subsequent acceleration of the pairs at the pul-sar wind termination shock, PWNe could produce a moresignificant contribution of primary cosmic ray positrons.This work was supported by a NASA AstrophysicsTheory grant and a Fermi
Cycle-5 guest investigatorgrant.APPENDIX A. ALGORITHMS FOR SEMI-ANALYTICAL CALCULATION OF CASCADE MULTIPLICITY
Here we show pseudo-codes of algorithms used to compute cascade multiplicity. For calculation of χ a ( (cid:15) γ , B, ρ c ) wecomputed and stored a table of 1 /χ a values for a uniformly divided grid 77 × ×
20 in log (cid:15) γ × log B × log ρ c space,and then used cubic piece-polynomial interpolation to get χ a for parameter values required by expressions used in thealgorithms.Algorithm 1 computes total number of particles produced in synchrotron cascade initiated by a single primary photonwith the energy (cid:15) γ , physical processes are described in § e phot and n phot are the number and the energy ofsynchrotron photons emitted by particles of the current generation, e phot is the energy of escaping photons, and n is the total number of particles produced in all previous generations.Algorithm 2 computes the total multiplicity κ of a CR-synchrotron cascade according to eq. (23). e part is theenergy of the primary particle at the current distance s i ; e phot is the energy and n phot is the number of CR photonsemitted by the primary particle at the current distance, Nsyn is the number of particles generated in synchrotroncascades initiated by the primary particle at the current distance. Integration over the distance is done with thesimple trapezoidal rule. B. TRANSITION TO RADIATION REACTION LIMITED REGIME
Radiation reaction begins playing an important role in the dynamics of particle acceleration if particle energy lossesper unit time become comparable to the work done on the particle by the accelerating electric field. For curvature6
Timokhin & Harding
Data : (cid:15) γ – energy of primary photon, s esc – mfp of escaping photon Result : N syn – total number of particles produced in synchrotron cascade initiated by a photon with the energy (cid:15) γ Function N syn ( (cid:15) γ , s esc ) : e esc ← (cid:15) γ, esc ( s esc ) // eq. (10) e phot ← (cid:15) γ n phot ← n ← while e phot ≥ e esc do n ← n + 2 n photn phot ← n phot n syn ( e phot ) // eq. (15) e phot ← (cid:15) ( i +1) γ ( e phot ) // eq. (16) endreturn n end Algorithm 1:
Multiplicity of synchrotron cascade
Data : (cid:15) ± – energy of the primary particle, s cr – characteristic size of the cascade, s esc – mfp of escaping photons Result : total number of pairs produced by the primary particle with the energy (cid:15) ± begin divide [0 , s cr ] in subintervals s , . . . , s i max Nsyn last ← n ← for i ← to i max do // CR radiation at distance s i e part ← (cid:15) ± ( (cid:15) ± , s i ) // eq. (19) e phot ← (cid:15) γ, cr ( e part ) // eq. (21) n phot ← n cr ( e part ) // eq. (22)// synchrotron cascade multiplicity Nsyn ← n phot N syn ( e phot , s esc ) // algorithm 1// trapezoidal rule n ← n + 0 . Nsyn + Nsyn last )( s i − s i − ) Nsyn last ← Nsyn endreturn n end Algorithm 2:
Multiplicity of CR-synchrotron cascaderadiation the condition for applicability of the free acceleration regime is (see eq. (17)) eEc (cid:38) e c (cid:18) c ρ c (cid:19) (cid:15) ± . (B1)The electric field E grows linearly with the distance l ± traveled by the particle in the gap, eq. (27); the particle energy (cid:15) ± increases as l ± , eq. (32), and at some distance the condition (B1) is violated. However, if condition (B1) holdsup to the distance l e ± where the particle emits “gap-terminating” photons, then the gap length and the final particleenergies can be well described by the free acceleration regime, neglecting radiation energy losses. Using eq. (27) for E ,eq. (32) for (cid:15) ± , and substituting l e ± , gap for l ± from the eq. (39) inequality (B1) becomes the condition on the magneticfield strength B > α f χ a B q (cid:39) . × − B q = 1 . × G . (B2)In the second step we used the value χ a = 1 /
7. The radius of curvature ρ c and pulsar period P cancel out, except fora weak dependence of χ a on ρ c . We see that for most non-millisecond pulsars radiation reaction can be neglected. C. LIMIT ON 1-D APPROXIMATION
The one-dimensional approximation for particle acceleration in the gaps works well if the length of the gap is muchsmaller that the width of the polar cap. As the length of the gap is ∼ l gap , we can set the formallimit on 1-D approximation as l gap < r pc , (C1) air multiplicity in young pulsars r pc = (cid:112) πR ns /P c is the polar cap radius. Using eq. (37), (39) for l gap and l e ± we get the following limits onpulsar parameters for applicability of the 1-D approximation B / P − / > (cid:18) c πR ns (cid:19) / (cid:32) χ a B q λ c c π (cid:33) / ξ − / j ρ / . (C2)The pulsar magnetic field is usually estimated assuming magnetodipolar energy losses from the values of period P andperiod derivative ˙ P as B = 3 . × (cid:112) P ˙ P . Expressing B in this way and putting the numerical values for physicalconstants, we get from eq. (C2) the following condition for applicability of 1-D approximation in terms of P and ˙ P ˙ P > . × − χ / a ξ − / j ρ c P / (cid:39) . × − ρ c, 7 P / , (C3)in the last step we used values ξ j = 2 and χ a = 1 /
7. For cascades along dipole magnetic field lines at the edge of thepolar cap with the radius of curvature ρ c = 43 R ns θ pc ≈ . × √ P cm , (C4)where θ pc = (cid:112) πR ns /cP is the colatitude of the polar cap edge, condition (C3) takes the form˙ P > × − P / . (C5)REFERENCES(C5)REFERENCES